Quick rundown: depending on your (hidden and unreadable) ELO, you'll either have 1v1 or 2v2 at the bottom lane, and the remaining 3 (or 4) go top. Bot lane should be able to push the minion wave without dying easily, while top should adopt a teamfight composition. (Initiator, AD, AP, and an extra of any). Fights generally happen in 2s and 3s, so everyone should be able to at least escape a 1v2 situation ideally. The basic plan is to hold the two points closest to your Nexus along with the top point, but don't be afraid to break away and either gank the enemy bot laner or back-cap enemy mid point if top is too hard to take. You need to hold 3 points, doesn't matter which in the long run.
As for champions, tanky damage-dealers are your best bet. Darius, Jax, Riven, Brand, Ryze, and Wukong are a few that come to mind. Since you're gonna be stuck in fights no matter what, you want to be able to take a burst and not instantly explode, but still do enough damage to fight back. Expect to die, though. A lot. Just don't do it pointlessly and you'll do alright.
If you want the big picture, the guide linked in the post above mine explains all the details far better than I just did, but will probably take you forever to read. :P Well worth reading for competitive Dominion though.