26. Kirby
I just simply do not think Kirby is better than anyone else overall. Bair is the only great thing he has. His recovery is the easiest to edge guard in the game, and no good grab combos either. At least Uair is strong. Some may disagree but I cannot think of a worse character than Kirby.
 

25. Zelda
A very overrated character; all she has is a 4 frame B air really, and a 4 frame down smash. Sheik can even Fair her shield then do a grab before Zelda can shield grab which is just horrible. Just camp her, and don't get hit by Bair, and she really cannot do anything. She is the most overrated character in the game by far in my opinion.

24. Ness
Lack of range and a bad recovery make him pretty awful. His grab combos can at least be decent with Uair strings or throwing them off the stage. Fair is also okay and double jump cancel Bairs can be good. Ness's lack of a range is a real problem. Ness could be higher than 24th, but he is a lot better than Zelda and Kirby. Those 2 characters are by far the worst.

23. Bowser
Very limited and gimmicky. Up B out of shield is his best tool. This is the best Bowser video by far, but only works because of lack of knowledge of the matchup

22. Game & Watch
The worst shield in the game, a 2 frame up B OOS (one frame up B but 2 frames if OOS) but that's all he has. Fair is extremely good and his planking can be decent if you don't wavedash out of shield to punish it.

21. Roy
Very bad combo weight and gets chain grabbed or death comboed by many characters. Otherwise he's still Marth just with worse moves but he's not THAT bad due to being so similar to Marth.

20. Mewtwo
Mewtwo gets camped really hard, but has a good edge guard ability with Back air. His back throw can also lead to Up throw combos, up throw kill, back throw to edge guard, or back throw kills. However all you have to do is camp him and stay center stage and he cannot do anything. Good players will combo him very hard because he is light weight but also large and you can even hit his tail.

19. Pichu
A very underrated character, his Nair strings alone (which only have 6 frames landing lag which is the smallest in the game) is his staple bread and butter and once you hit one you can continue a string of hits. 

He can also up throw up smash or up throw chain grab fast fallers. Pretty much to use Pichu, you use Nair in various ways for the most part. His forward roll and shield are both top tier. I think Pichu is underrated, even though maneuvering around and Nairing is his main thing it can be very effective in many situations just from that one move.

18. Link
Young Link has more good matchups than Link in my opinion, but Link's stronger and longer sword can make him pretty good at a lot of things. Lack of mobility can be quite an issue making him not very good vs any of the good characters.

17. Young Link
Pretty much only good for Uair strings (you can do up throw to Uairs on many characters such as Sheik for example) and camping Jigglypuff.

16. Yoshi
Amsa proved how good Yoshi could be with his large combos, the super armor from double jumping, the Uair strings, and the powershielding.

15. Donkey Kong
DK's up throw to Uair strings are his bread and butter game, and he is able to do many large combos by using cargo, up throw, into Uair strings. This is good against many characters, however, this is all he really has going for him. I do not believe there is a great top level DK to show that he can be the 15th best character. I may be very slightly overrating DK because without grab (and Bair) he is pretty horrible.
 

14. Ganondorf 
A very strong, and very easy to use character. However, I believe Ganondorf is extremely, and I mean EXTREMELY, overrated. When you aren't on Yoshi's Story, he has a very hard time killing or approaching. 

All forms of projectile camping or camping in general can beat him, and a good player will gimp his recovery very often. It is possible that Ganondorf is better than Mario, but I do not think he is better than any of the characters above him. He is very easy to use on the most basic level, but the more high level play you get, the more his weaknesses really shine out.

13. Mario
It is hard to rank Mario exactly, since he is so similar to Dr. Mario, however, due to a far worse Bair and Cape (super sheet; which does not go as high or low vertically which is very important, but goes better horizontally), his edge guarding becomes drastically worse. 

One of his best moves, down smash, is also solidly weaker than Dr. Mario's down smash. Fireballs are also a lot worse than Pills/Megavitamins are. He can chain grab for longer %s however, and he does have a cape for better recovery (although Dr. Mario's horizontal aerial moving distance is a bit greater). I believe Mario is around 13th in the game.

12. Samus 
Samus's wavedash back into a down smash, down tilt, grab, or forward tilt, are some of the staple things about her. She is also good vs Falco because of her Screw attack (up B) out of shield. I could see Samus being better than Luigi because of her strong match up vs Space animals (Fox/Falco) but that would be the only reason.
 

11. Luigi
Luigi's wavedash allows him to move across stages very quickly in a way that you often cannot react to it if he does it properly. Abate will do things such as dash away from you, wavedash at you, and then grab you into a down throw, and before you know it, you are already thrown, without any DI (directional influence), and then he combos you hard off of that. 

His down smash at low % combos very, very hard into more down smashes or other aerials which work on all fast fallers and even Sheik. Marth is floaty enough to escape these combos quite easily. Luigi's fluid movement and decent combo game make him okay. He has a 3 frame 15% nair that combos into other moves which is very good to use, however, you can bait Luigis into this. He is also easy to shinespike or gimp with many characters.

10. Pikachu
I believe Pikachu is the 10th best character. Pikachu can chain grab the fast-fallers on FD, back throw to Nair or Uair or Fair+Uair to gimp almost any character, do a short hop Nair that combos and then land behind you so that you cannot punish him, he has a very powerful up smash and up throw combos into up smash often. 

He is a good overall character. I could see him possibly being better than Dr. Mario because of how quickly and easily he can jump off stage to gimp the opponent and still recover quite easily if he misses.

9. Dr. Mario
He can chain grab basically everybody with up throw and down throw, edge guard amazingly with Bair and/or cape, pills are a decent projectile, and crouch cancel down smash is very good. These are all good tools, but his lack of range is a huge weakness.

8. Captain Falcon
The "borderline" character. Falcon is either the worst viable or the best un-viable character. He beats most characters with simple up throw then many Uairs, and he can 0-death very quickly, however they are not usually guaranteed and often require guesses/reads which makes Falcon at top level rather inconsistent. Being inconsistent is bad.

7. Ice Climbers
Ice Climbers have a far better match up vs the top tiers than Captain Falcon does. Fox/Falco/Sheik/Marth (debatable)/Puff (by far). I think it is wrong of the community to assume Falcon is better than Ice climbers. Ice climbers can 0-death you very quickly with one grab, but they rely on that a lot. Ice climbers are also better vs the top 4 characters in the game than Captain Falcon is.

6. Peach
Peach has about a 50/50 match up vs. Falco, but she loses to every single other character above her on the tier list. She also loses to Captain Falcon slightly, although that is debatable. She hard counters Ice Climbers.

Armada has won many nationals with Peach, such as Genesis 2, Apex 2013, and Apex 2014, so we know Peach is at least pretty good, but I think the above 5 characters are better. Peach is close to as good as Marth.

5. Marth
Despite the fact that Armada has won more Peach tournaments than me or any other Marth have won, I still think Marth is better than Peach (except Peach is better vs Falco). Marth is just extremely good overall, but he has problems KOing at high %s and also has no good out of shield options. He loses to the 4 characters above him except on Final Destination. 

His recovery is pretty bad and easy to cover all of his options in many situations, and he gets 0-deathed pretty hard. However, he can also 0-death you very quickly. He is the worst of the top tiers, but better than any high tiers. If Final Destination was the only stage in Melee, then Marth would be hands down the best character overall, no contest. He does become much worse on other stages however.

4. Sheik
Despite the fact that Sheik loses to Fox/Falco/Puff/Ice Climbers, she is SO good vs. all of the other characters that I believe she deserves the number 4 spot. There was one point in time that I thought she was the best character in the game, but I found out that she had many limitations (such as recovery or lack of good approach) that the above characters do not have. 

In teams, Sheik is by far one of the best characters, has the best edge game tied with Jigglypuff, and the best edge guarding period. Sheik is an extremely good character, I just do not currently think that she is better than any of the top 3 characters. I could see her being better than Jigglypuff possibly, but currently I do not hold that opinion, but I could see it being true. 

3.Jigglypuff

Overall Jigglypuff can go for low risk, high-reward all the time, which makes her very powerful. Her weight allows her to escape combos very easily and DI off stage whenever needed and still be okay. She is a very underrated character; and I am not just talking here, I actually use Jigglypuff quite a bit in tournament recently (despite my total play time with her DRASTICALLY lower than most of my other top tier characters), but mostly for teams because she excels here.

2.Fox

Pretty self-explanatory, Fox is very similar to Falco and his tools make him some of the best of the best. Not much to say here that is not already obvious. Fox is also better in teams than Falco and if you average teams with singles, then Fox is the best character. 
 

1.Falco

I have believed that Falco was the best for a very long time since 2005 to the present based on numerous reasons.

Results is one reason. One reason is that not only do I lose to this character personally (despite being the best Marth and Sheik against Falco in the world in both matchups), but I see various Falcos always winning tournaments. Some notable ones that win majors all the time were PC Chris, Mango, and Dr PeePee. Various Falcos have won more major tournaments than any other character in the past 9 years, and I believe that really says something about how good the character is. Unlike Fox, he is easier to play consistently at a top level, making for higher levels of consistency. 

Guaranteed combos are a lot more common with Falco than Fox. You cannot DI out of his combos as easily (usually not at all in many top level scenarios) and in many cases there is nothing you can do at all in many situations except for eating a large guaranteed combo. Fox, for instance, you can DI or smash DI his Dair to escape quickly, or DI down when he does Nair combos and then mix up your tech rolling options to have a 50/50 to escape the combo. Falco's combos are more guaranteed; his short hop Bair both autocancels and has more range, and Dair has much more range, he has a guaranteed spike (Dair) that you cannot meteor cancel, sets up for massive guaranteed combos on platforms (or even on FD is inescapable), and outprioritizes most moves without even trading. Once a Dair hits you, you are often going to eat a death combo in many situatons, or at least a solid punishment.

You can more easily react to edge guarding Fox than Falco, because when you jump off the stage, you can visually see if Fox chooses to use an Illusion or a Firefox. With Falco, because his phantasm (illusion) is a few frames faster, you cannot simply jump off stage to react to both options; instead, you have to pick one option and cover that one, and you will be unable to cover the other option usually. Fox's illusion is slow enough that you can often react to both options in many situations. (Falco's illusion also spikes you, making it more threatening should you get hit by it).

Falco also has solidly more range in almost all of his moves, and and he is heavier both horizontally (weight) and vertically (mostly terminal falling speed). The range advantage is pretty important because almost ALL of his moves have solidly more range than Fox.

Tech chasing is far more dangerous against Falco than Fox, because of 2 reasons. One reason is that his tech roll goes massively further than Fox's, meaning that if Sheik down throws Falco, and is trying to react in a way that will cover all the options, and Falco tech rolls away, he can usually spot dodge, jab, or dash away before the Sheik can regrab or dash attack him. Combine this with reason 2, which is tech in place + Shine. 

You do not have to do this all the time (you can jab or buffer roll for example), but doing this about half the time is a very good mixup because it is very easy to combo off of the shine. Should you waveshine towards the enemy, you get a free huge combo (from jumping above them and Dairing at the last second before they escape the hitstun). 

Even if you miss the waveshine and just do a regular shine, if they do not DI away (which will happen often since the Sheik is not expecting to be hit) you still get an easy combo. With Fox, you are less likely to get a waveshine combo in my experience (by a lot) than you are a Falco combo. But you can also react to all the options with Fox on tech chasing because of how little range his tech roll goes. You can have a delayed reaction on Fox, and still be able to dash grab his tech roll, where as if you have the same reaction time on Falco, you will miss him often.

Perhaps the biggest thing of all, however, is lasers. If you do not powershield consistently, these lock you down really hard. (And nobody has a higher than 50% powershield rate; and even when you do powershield, if the Falco just camp it does only 1% (half damage) back to Falco. Furthermore, if Falco decides to dash away (like he should usually be doing) after a Short Hop Laser, it automatically smash DIs away the laser away from Falco so that Marth's powershield grab from neutral stance won't even reach Falco. This means the only good approach to Falco is to run and hope you get the powershield and then lead it into a Grab or Aerial. Due to the fact that the powershield window is so incredibly small (and you may light shield by accident sometimes even if you do not think you are), this makes this type of approach rather inconsistent with most controllers and characters. Falco can also condition you with lasers and do combos like Mango's Laser -> F tilt (always works) or Laser -> F smash (works if you hit the opponent out of the air). You can be across the other side of the stage and still use the laser to get a guaranteed combo on the opponent in many situations. This also includes a jab-reset, into a forward smash, from across the entire stage on a missed tech (which happens often in my experience). In situations where you do not have a guaranteed combo, you can still condition them or put them in a situation where you can be safe (such as get in their face and dash away) but close enough that they have to use their human reaction time (12-15 frames) to decide that they probably have to take action, and then whiff a move and often be punished for it.

Shine-grab also grabs before you (any character) can shield grab, and laser-grab also grabs before you can shield grab. If you do not have a quick shine OOS or nair oos, then you must roll away or not be in that scenario to avoid this. You can also spot dodge but that must be timed perfectly and should Falco do anything else, you are going to get punished hard. Another pressure mixup is to rising Nair after shine (happens before anyone except maybe sheik can nair OOS), and pull backwards out of shield grab range (not even Marth can shield grab it). Fox can do this too however (and Fox's nair and pull back on shield after shine, is BETTER than Falco's slightly), but Falco's shine grab is better.

Why is Falco's shine-grab better? If Fox's shine hits, Fox gets absolutely nothing out of it except 4-5 damage and a neutral stance. If Falco's shine hits, it leads to a massive combo even if the grab part of the shine-grab whiffs. Most situations you also will not be DIing the shine, meaning you can do a FH Uair (or any aerial) and then do a combo from there. Even if you are DIing away the shine, Falco can always Short hop laser towards you into an F tilt or F smash. Reason number 2 that Falco's shine grab is better is because Falco's shine has 1 more frame of shield stun advantage than Fox's. Fox's shine to aerial is 1 frame superior to Falco's because Fox's jump is 2 frames faster than Falco's (in air frame 4 instead of 6), however since you do not need to be in the air to initiate the shine-grab, this makes Falco's shine-grab +1 better than Fox's, in addition to a shine-combo should the grab miss. Fox's shine grab can be better in a situation where you can up throw Uair to KO the opponent. Most situations it is not better however.

I have multiple reasons for thinking Falco is better than Fox, which many (from the top of my head) are listed above just from the top of my head. Despite Fox being more common as a secondary or among mid or high level players, the tournament results in the past 9 years as well as consistency and the theorycraft i listed above all support Falco as being the best character. I also personally believe Falco 60:40s Marth and Sheik, where as Fox 55:45s Marth and Sheik.