I'll give it a shot by explaining dominion and then try to give some tips in it aswell.
Dominion is a capture the point style map, what this requires to win is having access to 3 points (or more) at all times. The five points can be identified as followed: The Windmill(Top), The Refinery (Blue Left Top), The Quarry (Blue Left Bottom), The Drill (Purple Top Right), The Boneyard (Purple Right Bottom). The five capture points when taken will act as turrets similar to summoners rift so don't think you can take them easy, if there's a champion on it to disrupt your channeling of the point then you'll find the point will attack you too.
Generally most setups have people go four people head towards windmill one person goes to bot (one however from top will fall behind to capture the refinery). Strong champions ofcourse are the ones with map mobility (Kassadin, Shaco, Rammus(Note these can fall off easily)) or general burst champions like akali, master yi, wukong, lee sin. You won't find many players who actually play the support role on this map.
Quests: When a point gets pinged and you either see a sword or a shield over it this becomes a quest, you must protect your point while actively taking theirs, the quest basically gives a +15% damage increase to your entire team and deals more damage to the enemies nexus. However if they are defending it vigilantly don't continue attacking see if they've left an area undefended and take the chance.
Dominion Buffs: There are three types of buffs in dominion you have the speed buff, the health relic and the greater relics(These can be found in the center of the map). The greater relics are ignored in some games due to the game can be so fast paced you're always on the attack and defend, however gaining this buff can help dramatically. Especially on a good carry. What the greater relics do is they grant you a lightning shield buff which will absorb a certain amount of damage (recharges 15seconds out of combat) and give you a chance for your attacks to proc a lightning attack. This can help turn the tide of some fights.
Summoner Spells: Now this section may be a little beefy as it'll cover a lot. Generally it comes down to knowing when to attack and when to defend if you find yourself more a defensive player strong picks could either be : Garrison & Exhaust, the reason being is there's always that one burst champion that always thinks they'll get away with attacking you on your point, garrison will improve the points damage and regen. Exhaust will help lower there damage output and speed so it's more than likely you'll shut them down here.
Players that enjoy pushing the choices could be Promote & Exhaust (Now I've just said exhaust is a defensive spell, however it's also great for attacking) Promote will basically make your large minion gain more health & damage, this helps push a lane more effectively and it helps you take a point easier too.
Recommended Champions: Attackers : Akali, Kassadin, Shaco, Wukong, Lee Sin, Volibear, Jax (Dodge Runes Required IF you still have them), Tryndamere, Master Yi & Fizz. Defenders: Heimerdinger, Rammus, Amumu, Gailio, Garen. Please note these are recommended I know many other people will say X is strong but these are basically for pick up and very simple to use.
Generally that's all I can think of right now i'll add more to it when I can.
/Czeldabolg
ratio, items such a lifesteal and spellvamp help alot to keep up your health if needing to defend one of the 5 points. If your wanting to go for top player make sure you kill minions i think its atleast 2extra points per a minion in a game