The title says it all.

Kha'zix/Kassadin/Shaco

Role: Jungler
Champs: Akali/Xin Zhao/Darius

A jungle that gets really strong when fed, Jax, Xin, Jarvan, Hecarim

issue with this post:
likely hood you will actually be allowed to play that role
How good are you with even one champion
How is your map awareness
How well do you know other champions

But top or JG are the two best roles to carry yourself out.
hard part is actually getting to play those roles.
Garen top <- noob pick but easy snowballer or tank
Xin , lee , kha , shaco , vi <-- Mobile and great if you can continuously pressure lanes

Mid: Lux, Kass
Jung: Xin, Vi, Hec

I hate to be the guy that says all roles...But all roles can carry yourself out. If you can have 100% comfort in any spot and know you caneither feed your carry or carry hard yourself. You can do it.
But if you don't wanna hear any of that.

I think a good mid can carry any game (assuming you win lane and roam to bot to help feed your carry.

Thats how I managed to climb from Bronze V to almost silver V (bronze I atm)

Try Akali. People can't handle her burst in lower divisions. Same goes for Draven. Give it a go.

I recently purchased singed. Now, I am not good with him or really any champion for that matter.. I am in bronze 3, after all..That said, my feeling is that singed would work extremely well, because Ive noticed that people seem to follow singed's poison trail from full health until they die..it is quite amusing, actually..and always frustrating when there is a singed on the other team and no one is paying any attention to my ongoing pleas for my team to stop chasing him.