Well, I am a noob and haven't seen every champion in a fight...
Melee Carry - Pantheon
When fighting against him you have to constantly be aware or he will use his ultimate, then stun you and do massive damage. He even reduces damage with his passive giving him a bit more survivability.
Ranged Carry - Teemo
Turn on poison and blind, right click. You win. Good escape too.
Nuker - Ryze
Does anything need to be said? Rune Prison then destroy.
Support - Heimerdinger
He can do decent damage, but his turrets can completely control the flow of a lane at ability level 5.
Tank - Singed or Malphite
Malphite can initiate great, and take a lot of damage. Singed can't take much damage for a tank, but he can determine where an enemy goes. Chasing him or his team while he is with the team is stupid, his placeable slow is great for escaping or stopping a few enemies from escaping, and fling is the death of enemies in team battles, run up to their carry and fling them into your team and they all jump on him, because of that people stay away from singed, so the enemy team better have a lot of stuns. His ultimate makes him do some decent melee damage and increased his health regen and tanking ability and speed, what more do you want?