Strongest Mid Laners

Zed [Damage/Outplay]: In terms of team utility, Zed essentially has none. He is a pure damage and outplaying machine, and efficient management of poke, clones, and all-in timing is crucial to pull off that sweet ultimate + nuke + fade away back into the shadows. However, pulling off amazing plays with him is probably one of the most satisfying things you'll ever do in League of Legends.

Katarina [Damage]: In terms of raw damage potential Katarina is bar none, the strongest snowball champion in the game. The second she kills or gets an assist off one champion, her damage effectively increases by an instant 50~100%. This combined with the fact that 75% of her skills are AOE means that she WILL get a reset and your team WILL die.

Leblanc [Damage/Utility]: A good Leblanc is absolutely indispensable. Surprisingly, it's her ability to poke with distortion that separates a good Leblanc from a bad one (there are no mediocre Leblancs, but plenty of bad ones). She is an extremely high risk, high reward champion, and her most popular skill order puts her right in the middle of the fray to maximize her AOE damage output. However, with good timing and skill management, she can also lure out dangerous spells via her poke before she goes all-in to burst down a squishy.

Cassiopeia [Damage]: Although Cassiopeia does provide some utility, the reason why she's got an insane skill to reward ratio is simply because of her damage potential from landing her noxious blasts. If she can manage her blasts combined with her Twin Fangs, her cheesing power is through the roof. That being said, her mechanics and short cooldowns essentially make her attack almost as fast as an AD carry (with spells).

Syndra [Damage/Utility]: With her high range, short cooldown AOE stun, Syndra has the potential to be one of the most devastating mid laners in the game. If a Syndra hits with her nuke on every cooldown, almost any champion will die before even hitting the level two mark. Meanwhile, she scales well into late game and has the ability to delete a single player at almost any point in the game with a point and click ultimate.

Honorable Mentions: Ahri, Akali, Azir, Fizz, Karthus, Orianna, Yasuo. These champions are tried and true, and often have the ability to dominate solo queue by themselves. However, at the moment their damage output isn't quite as strong as others in their pool in the current patch. I would keep them tight on the radar though, as they're strong contenders in their own right.

Strongest Top Laners

Azir [Utility]: Azir's kit offers an insane amount of damage and utility if he can position himself and his soldiers correctly while managing cooldowns. Late game, his soldiers have the ability to kill squishies almost instantly after spawning and charging. He also offers a lot of champion displacement, and can block a lot of "jumping" skillshots (eg. Lee Sin or Kha'zix) via his ultimate.

Gnar [Utility]: Gnar's kit offers a ton in terms of kiting potential in ranged form as well as CC in his mega Gnar form. If players are able to understand the potentials of both forms and engage/disengage while managing his rage bar, he's definitely one of the strongest champions on the field at the moment.

Jayce [Damage]: Jayce's damage output is through the roof in terms of poke potential, and can safely take down enemy champions from far outside the fog of war. Not only that, his AOE dueling power is immense against tightly clumped up melee champions (read: supports trying to peel). A fed Jayce can clean up an entire team by himself with one quick dunk or whittle them down slowly from sniper range.

Lissandra [Damage/Utility]: Lissandra's ability to outplay via her untargetability combined with her long distance initiation/mind game power gives her extremely high utility. This, in sync with her strong AOE damage output means that she can shred down targets extremely efficiently when combined with her teammates. 

Rengar [Damage]: Rengar is basically a split-pushing god once he gets ahead top lane. His teamfight power isn't nearly as good since he'll have to rely on his pick power, but in a 1v1 or even a 1v2 situation, his ability to jump through walls and gap close from every bush gives him insane cheesing power. As a manaless sustaining champion, it's also very difficult to force him out of lane if you fall behind, and 1v1 it is very difficult to recover from an early game advantage.  

Honorable Mentions: Riven. Nidalee

These two champions have great utility and definitely have their perks. Riven with her insane ability to outplay via cooldown management, and Nidalee with her ability to jump through walls. However, in their current state I don't think that they give quite as much as the 5 champions above. 

Jungle

In terms of junglers, skillcaps champion-wise are fairly low and many are highly effective in jungle. However, there are a couple stand-out ones who do require higher skill to play effectively.

Fiddlesticks [Damage]: Few junglers (or champions for that matter) have the ability to make an entire team run or die at almost any point in the game. Fiddle is one such champion, and through his ultimate, has the ability to change the tides of a game by himself. However, because of his weak constitution, he needs to be careful in terms of when and how he engages. Getting interrupted or instantly bursted down before using Zhonyas is enough to make bad Fiddles completely ineffective.

Jarvan IV [Utility/Damage]: Jarvan IV currently stands at the apex of junglers in Patch 4.21 and rightly so. If mastered, we have here a champion with two built-in flashes, an AOE knockup, tons of passive stats from his skills, and one of the most unique forms of CC in the game through his ultimate.

Lee Sin [Utility]: Some of you may be surprised that Lee's not higher up considering how much free press he gets from pros. Lee's ability to dodge skillshots and make insane plays is highly touted worldwide and rightly so. He can do it. Although nerfed many many times in the past, his kit still remains highly versatile and gives players the ability to make crazy plays.

Master Yi [Damage/Outplay]: Master Yi's main draw as a high skillcap champion lies in his alpha strike, which effectively negates whatever was just thrown at him. Additionally, Master Yi's renowned base backdooring abilities make him one of the best come-from-behind; I'll-win-this-myself; and feed-game-win-game champions in the game. 

Strongest Supports

Janna [Utility]: Janna is close to being indisputably the best support in the game in terms of sheer utility that she provides by herself. Through both her tornado and her monsoon (ultimate), Janna has the ability to block a massive number of champion "jump" attacks mid-air, including but not limited to the following:

Lee Sin, Akali, Kha'zix, Tristana, Corki, Lucian, Jarvan, Ahri, Fizz, Pantheon, Shen, Wukong, Xin Zhao, Vi, Fiora, Amumu...the list goes on and on.

Not only that, but she provides nearby teammates the ability to simply be faster than enemy champions whether it's for running away or for chasing down. This combined with the ability for her shield to deny death and also give an insanely strong damage buff cements her as a top tier support.

Thresh [Utility]: Where Janna is the queen of preventing plays from happening, Thresh is the offensive version, and king of making plays happen. His lantern and hook make him essentially the best support initiation in the game, as he can bring a friend with him during his initiation as a free gap closer. Additionally, all the champions listed above that Janna can block Thresh can also block with his flay (although with a shorter range/window of opportunity).

Blitzcrank [Utility]: Where Janna and Thresh keep teams ahead of the game, Blitzcrank has the ability to take away leads from even careful enemy teams. With his hook, he has the ability to create advantageous teamfights without needing to get close enough to the rest of the enemy team. This, combined with his speedboost makes it so that he can get the job done.

Strongest Marksmen (AD Carries)

Ezreal [Damage/Utility]: Technically speaking, if you were to hit every one of his Qs on enemy champions every time it came off CD except when they flashed or otherwise dodged it on purpose, Ezreal would be the most broken AD carry in the game, bar none. Luckily, most people seem to only be able to hit around 20%, if that.

Lucian [Damage/Utility]: It's no surprise to most that Lucian would be on this list, as his ability to auto attack reset, dash around, and deal massive damage from afar make him an ideal AD carry. While he's much more balanced with the slew of nerfs Riot hit him with, he's still got high potential.

Sivir [Damage/Utility]: Where Ezreal and Lucian feel smooth to play with the complicated mechanics, Sivir falls short in the quality of life department, and often feels the pain of her somewhat slow boomerang and sometimes clunky auto attack reset animation from her W. However, when mastered her burst damage is absolutely outrageous, and can easily chunk someone like Sona down to half in a single combo at level 3.

Draven [Damage]: Draven has the strongest level 1 burst damage in the game bar none. 1v1, he will beat everybody. If he's able to all-in level 1 with a creep advantage, he essentially wins his lane then and there, provided he catches all his axes.

Vayne [Damage/Utility]: I guess no list would be complete without Vayne. As a short range AD carry with insane late game but mediocre early game, Vayne happens to be the favorite of cheaters worldwide, and for good reason too. As a champion with the ability to dodge skillshots on the fly, a geometrically based stun, and auto attack animation cancelling as her core driver, when mastered, her utility and damage is indeed absolutely through the roof.