In terms of gameplay/ carries out their archetype best. Not in terms of balance.
As annoying as he may be to face, I would say 
His kit is very smooth, and he really carries out the feel of a melee master with any weapon.
I might be a bit biased on this one, but I think Hecarim is one of our better designed characters in terms of matching their archetype.
His high mobility and ability to crash into enemies lines (both via E and R) really makes me feel like terrifying undead cavalry. Also running down their ADC as they flee across the map is alarmingly satisfying...
A few other well done archetypes off the top of my head (as personal opinion without justification :P):
I've got to give it to 
on two points:
Nobody fulfills the 'burst mage' archetype like the champ who can 100-0 a tank, late-game. Who's entire point is to turn the fight into a 4v5.
I think his identity fits in really well with his playstyle. In lane, since he wants to farm q, Veigar has no real lane presence. But he wants you to "know that if the tables were turned, [he] would show you no mercy!" And then you get to lategame and pop someone, and know that the tables have been turned.
Rammus!
Oh my lord this champion is so well done. You take one look at his splash art and think - roving armored tank machine. Then you pick him up and what is he? A powerballing team-tanking gank monster. Every part of his kit feeds this fantasy - his powerball is perfect for the "armordillo" concept.
i love the synergy between taunt, ult, and def ball curl. Again, with the armordillo concept it makes total sense that Rammus' primary way of fighting is to just ball up and let the enemy just get destroyed while they waste their time trying to break his shell. Even his passive is perfect- as a player you understand that this is a character designed to stack armor, and pretty much nothing else.
I personally believe that 
fits best into his archtype (self sacrificing support/super tank). He jumps INFRONT of allies and takes the damage for them. This perfectly fits into braums lore and his "attitude" (i.e. his quotes) in game.
The best IMO:
- I just love the way she feels/plays out. She's not my favorite overall, but I think Riot captured her feeling right.
- I think she was pretty well executed as a punching machine.
- She pretty much nails the concept of a poisonous snake-medusa lady.
So those are the best ones.
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To me the worst are:
- I can't tell if she's supposed to be a melee auto attacker or a Q spammer. Her last nerf left a lot of Eve players wondering what she is. So if you can't id her archetype she's got to be bad.
- I love Ahri's concept but I'm not sure what she is anymore. She's not mobile without her Ult, and even when she has it, it is limited. She's not really a mage now, but she's not an assassin either. So this is a case I can tell what she is supposed to be, but I can tell she's whiffing on all accounts aside from her visuals.
Jax does do the crazy damages quite well.
After a long thought and doing some scholarly tea sipping, I believe one has to specify what role the champion is fulfilling in order to judge how well they are designed. It's difficult to compare a Mage to an AD Carry or a Tank to a Support because they're roles, mechanics, playstyle, theme, and kits are all different. Here's what I think in terms of roles:
Best Designed Tank: 
I believe Malphite is the best designed tank because of his high armor scaling, great interrupt, CC, AoE and single target mixed damage, and overall tankiness. For a giant, sentient rock, Malphite fits the bill perfectly.
Best Designed Jungler: 
I believe Skarner is the best designed jungler because of his low cooldowns along with self reducing timers, quick clear time, strong, spammable, low cost AoE damage, high survivability with a shield, high mobility, great dueler and brawler, line-style slow, and the greatest form of CC with his suppression, Skarner makes a great bruiser in the jungle. He is a scorpion made of crystals!
Best Designed Mage: 
I believe Anivia is the best designed mage/caster because of her weak resistances, rebirthing opportunity, spammable CC, high burst damage, great interrupt and confusion inducing wall, AoE circle of wintry death, and good poke. Anivia players value timing and precision, making her a great mage. Her theme is also executed very well.
Best Designed Support: 
Although she may be weak as of right now, Sona is by far one of the best designed supports. She has great healing and sustain, spammable poke, mobility, and engage. She brings a lot of utility which is what a support should do.
Best Designed ADC: 
Quinn (and Valor) is the best designed ADC because of her strong poke and attack damage, AoE blind, scouting power, great reposition, and an amazing ultimate bundled with gap closers, blinds, and an AoE execute. Quinn, the falconer, makes great use of the theme.
Of course, these are my opinions and most people will not agree with me. But hey, thats all part of the discussion!
OP you're gonna have to clarify whether you mean who delivers on their archetype/fantasy the best or who's gameplay design is the best.
I thought you meant fantasy, but many people who responded seem to have gone the gameplay route.
In terms of delivering on a fantasy, my vote goes to Yasuo, Jinx, or Nautilus.
In terms of gameplay design, I think characters with clear strengths and weaknesses, like Kog'Maw and Braum are the healthiest.
Hmmmm, it's a tough choice, but I'd say Shyvana.
Her defining character trait (imho) is her anger at the world around her, and the rampaging that ensues because of it. Her Q gives her a great front-loaded momentum feel (more hits more Q's) and her W is perfect for the chasing and in-discriminatory destruction I feel like she would do. Not to mention her ult is just glorious in terms of going berserk. I'd even go as far to say she goes berserk better than Trynd or Olaf.
Support: 
Jungle: 
ADC: 
Top: 
Mid: 
Those are just my favourites in terms of each lane/starting place.
Role-wise, I have to say I love Voli and Malphite as tanks, and as much as I hate him in game, Talon is the best assassin ever. I don't play her, but Sona is a pretty solid support when played well, and Ziggs is the best AP carry by far.
Some people might argue with me and that's fine, but I really think Ziggs is the best champion Rito has put out. He doesn't have an innate burstiness (cough LB), so he takes some skill in fights, which IMO is more fun. Bouncing Bomb skill-shots and timing/placement of ults and satchels are key. People say he's useless without blue buff, but I just get Tear of the Goddess early, and farm up my mana with that. If I'm lacking mana mid to late game, I get Iceborn Gauntlet.
Beyond play-style, Ziggs has the best character/personality IMO. The crazy explosives-obsessed yordle aspect is awesome, and his care-free, oblivious nature is hilarious.
"Whaddya mean, it doesn't look safe?"
Maybe I'm biased, as I was quite a chemistry nerd in high school and still love explosions (the legal type, of course), but I just love everything about Ziggs. Rito, please never change him. :D
Serious talk:
best top
best mid the burst is real
reworked q 
best jungler
best support
best adc
Troll talk:
best top
best jungle
best mid
best adc
best support
Yolo talk:
best top
best jungle
best mid
best adc
best support
For champions that match a theme I would have to say Gangplank. He makes you feel like a pirate.
Cassiopeia is probably the best designed champion in the game imo. Her kit is just pure synergy (Q, W designed to apply poison for the E reset DPS) and her abilities are very strongly linked to the medusa snake lady thematics she has going.
Orianna (all her abilities come form the ball, not from her), Syndra (playing with her spheres is epic fun) and Nautilus (really gives the feel of the clunky titan even though everyone builds to remove the intended low AS/MS weaknesses) are also up there as the best designed holistically.
Honorable mention to Maokai, Jinx and Vi.
No one's perfect but these guys come the closest to me.
Jax is far from best designed. As it stands right now, he's very overpowered. He's like Lee Sin, except instead of bouncing back and forth, he dives in and pentakills the team with no risk. I feel with number tweaks to make Tank Jax less effective, like they did with Vi, Jax will be balanced. As for who I think the best designed champions are, here is my opinion
Best Tank
The cow has seen better days, but he's THE tank. He definitely embodies the idea of getting in your opponent's face and tossing them around. Braum is an honorable mention here.
Best ADC
People cry about Caitlyn being overpowered, but she is the "Renekton bar" for marksmen. She has solid damage, abilities, and escape options. Vayne is an honorable mention here.
Best Support
Do I really need to elaborate on this? Thresh is just an amazing support. He has a great kit for helping your team out. Braum is an honorable mention here.
Best Mage
Ziggs is great for very similar reasons as Caitlyn. Great escape options, and solid damage. Ahri is an honorable mention here.
Best Assassin
I consider Yasuo more of an assassin than a fighter. He has some great juking potential, and he has great sustain, poke, gap clear, and crowd control. Zed is an honorable mention here.
Best Jungler
He may not be meta anymore, but Udyr has some of the best jungle clear and sustain out of any champion. He can easily burst down champions, and despite his vulnerability to crowd control, he's easily one of the best carries in the game. Vi is an honorable mention here.
Best Bruiser
Jayce just has a fantastic kit. I'd say despite the clear cooldown and mana constraints on him, Jayce definitely embodies how a 6 ability champion with no ultimate can still be a ton of fun. Lee Sin is an honorable mention here.
Worst Champions
Worst Tank
Urgot is just... I don't even know what Urgot is. He's ranged, but he has lower attack range than Thresh. He's a tank, so why is he ranged? His ultimate just doesn't make sense for his kit, and honestly nothing else really does either.
Worst ADC
Cue the downvotes. Ashe just doesn't have anything that really makes her a worthwhile ADC. Her passive is atrocious, she doesn't have an escape, and she doesn't really have poke either, despite having formerly the longest range in the game.
Worst Support
This guy is not a support. He's a bruiser who's been overnerfed. His kit is very toxic in terms of the kind of gameplay in bot lane. Post 6 he'll start to take CS, and he just doesn't belong in bot lane in this meta.
Worst Mage
Anivia is extremely squishy and doesn't have the damage to back it up. She just all around isn't very fun or satisfying to play.
Worst Assassin
Her kit is just putrid. The only think assassin like about her is her passive, and it's the easiest stealth to reveal.
Worst Jungler
Rammus has some very strong ganks, but when it comes to sustain, he's just way too lacking. His Q is his bread and butter however, and there really isn't a use to it outside of the jungle.
Worst Bruiser
Gangplank has a serious identity crisis. He's supposed to be a duelist, but has no attack speed steroid. He's supposed to be a carry, but is massively outscaled by pretty much every duelist in the game. He's supposed to be great in teamfights, but his ultimate is just not good enough. GP is my favorite champion, but he's not fit for purpose. Jack of All Trades, Master of None.
Worst Champion
Alas poor Yorick. Yorick is the embodiment of how NOT to create a champion. He has no counters, yet he is completely useless past laning phase. He's not fun to play, nor play against.
I think they should be define by role, not position.
Tank: Malphite. His passive fits his tankiness, his slows and aa enhancers really make him a great champ for peeling or engaging.
Assassin: 
Talon. He loves to pick off fleeing week targets with his passive. Its such a burst combo that makes his ability to kill great, but if they dont die, hes screwed.
Team Mage: 
Orianna. A great aoe concept mage. I love it how shes so team orientated. She has the perfect ability to disrupt and do damage.
Burst Mage: Veigar. Hes such a great show of how to burst out one target really fast. His E doesnt fit as much since its an aoe stun, but it synergies with his W so well, i decided to but i anyway.
ADC: 
Shes such a perfect late game carry, she just synergies so well with late game items
Engage Support: 
Leona. She has great tool or lockdown and engage, while her numbers need tweaking, her ideas are the best
Sustain Support: Sona. No answer needed.
Peel Support: 
Zyra. Her kit has great means to deal damage while creating great opportunities for her adc
Assassin Jungle: 
Rengar. He has a great combination of damage, cc and sustain. His ult wraps it up to make him the perfect idea of an assassin.
Tank Jungle: Perfect ability to clear, sustain, and take a ton of damage. While in it he does small aoe and cc. He is the best ambush tank.
