I like highly mobile champs, champs with good engage/escape, and good at catching up to enemy champs.
So naturally, some of my favorites are: Master Yi, Lee Sin, Rammus, Akali, Riven, Teemo, Aatrox, Diana, etc..
I also enjoy playing tanky champs that can take alot of focus and still survive, such as Galio, Cho'Gath, Shen, etc...
So by looking at these categories, I find that Master Yi fits my preference the best and it's no suprise that I've played hundreds of games with him.
He is highly mobile with the best base movement speed in the game, has good engages/escapes with his Q and his R and can also catch up with enemy champs the same way, and his W is good for survivability or when he's being focused.
Everything I like.
I would suggest picking a Champ that mostly has all the things you want in a Champ Cutty.
For Akali, she has a good engage/escape with her ult, good burst with her Q+R+E or any other combo, and she has a slow/escape with her W.