My biggest weakness as far as champions go is definitely with tanks/fighters/junglers.
However, as I've been playing a lot of ARAM and Dominion I have found myself learning more and more the joys of melee. Riot for some perverse reason seems to like to give me tanky characters in ARAM more often than not. heh At first I was certain I would stink up the lane but instead I discovered that there is a certain awesomeness to playing melee in team fights.

Plus in Dominion my favorite character to play currently is Rammus. At first I picked him sort of at random because I knew nothing of how to play the map. My first Dominion game was not only awesome but after the game I had two really cool players one from each team help me understand the play and learn. It was really encouraging but now I'm just off topic about Dominion. lol

Anyway at first I was really surprised how fun Rammus was to play on that map. Now he is my 'go to' Dillo for that arena. (I still need a good second, cuz sometimes other people pick him first. :D )