Kayle and Master Yi are definitely the best cheapest hero. "The best soldiers are soldiers who can heal themselves."
Of course they're good not only because they can heal. Kayle can heal, push, support, gank, solo, and be not squishy (when you don't know what champ to play, choose Kayle and she'll do all the jobs for you). Master Yi is undoubtedly one of the best pushers and junglers in game, especially when you take 9/0/21 mastery and ghost + teleport (for multi-lane pushing) spell. And he also has good escape mechanism (Highlander, his ult).
Ashe? Yes, she's got stun and slow and ranged, but she's the squishiest cheapest champ ever. If your teammates are not doing well (i.e. most of your towers are destroyed) you won't do well either, period.
Nunu? Yes, he's the toughest cheapest champ. Well, that's all he is for, to tank and to gank. If your teammates are not doing well, you can't win the game. He also can't clean waves of creeps, sadly. Haha.
Sivir? Well, I prefer Sivir to Ashe. She is a pusher, and we win the game by destroying towers and Nexus. But she can't heal, so maybe she's not the best cheapest champ available.
Tristana? No, she's not cheap. She's FREE. And she can push very well. You might want to add her on your list, buddy.
- By "cheapest" I'm referring to champs with 260rp/450ip price.
- Why is heal so important? Well, it helps a lot in early game for laning. You never go wrong with keeping your HP up until you reach higher level. And of course it also helps in late game.