Riot games have put a special effort into having balanced champions for this game style as well. I don't believe there are any champions to avoid. However, the team composition changes a bit.
AP Carry
You will still need one, or else the other team will just build against AD and will own you as soon as they are a bit stuffed.
Tank / Off Tank
King in solo holding the bot lane, also very good in "tower diving" (if we can still call it so in Dominion).
Melee Fighters
They tend to be the heroes everyone picks as they are sort of kings on this map. They deal quite some damage and have high sustainability (so they don't need to Back often), and are often very good in 1v1 (which happens pretty often in Dominion).
Ranged AD Carry
More than one is too much, but not having one (to have more fighters) is a handicap. He deals crazy damage and can attack out of tower range or defend minion waves without exposing himself in front of the tower. From the games I've played or seen, having only melee champs is clearly a handicap.
Support
I rarely see them in Dominion and believe they are overlooked. Each time I've had one on my team, it was awesome to hold the windmill (we never had to back, support would always buff/heal us).
However, having a support requires good team play, because the champion won't do anything on its own (compared to fighters).
General
As XP gain is so fast in Dominion, teams are usually built with mid/late game champions. Riot tried to correct this in a previous patch by lowering the XP gain aura on the map, however, you still get to level 6 tremendously fast, so don't hesitate in taking weak early game champs (like Kassadin, useless before level 6).
Try having a hero or 2 (at least) with high mobility (Rammus, Kassadin, Nocturne, Twisted Fate, etc.) to do lane swaps and be unpredictable in your tower attacks.
Conclusion
All heroes are viable. It all depends on the team composition.
Good Akali, Irelia and Xin Zhao champions can however be freaking awesome with their ganking abilities and life steal.
From the League of Legends Wiki:
"Carry" is a term that is typically used to refer to the primary damage dealer / killer on a team. So you will usually see a carry have the highest number of kills on that team, by the end of the game. Carrys may often start with low effectiveness at early game and rise to power through the mid / late game. The carry is usually a character that your team wants to feed, you want him to farm and get the best gear possible so he can own.
Basically, a carry starts out slow but becomes the most powerful member of the team if played effectively. If you feed a carry, his eventual power is designed to be greater than it would have been if you'd fed a fighter instead.
Fighters are champions that have constant damage output and good survivability. They tend to last a little longer in team fights as opposed to an assassin, deal more damage than a support class, and has more consistent damage than a champions who nuke (usually AP champs). They'll become powerful if you feed them, but not as powerful as a carry.
If someone says they are playing a fighter as a carry, I would guess that they're basically wanting the team to feed him/her. It can be a viable strategy, but feeding a legitimate carry is usually an easier path to victory.
Best Answer
League of Legends is free to play game.
That means that if you bought it (without any additional in game stuff) you have been scammed. For example if you buy Collector's_Edition you will get champions (which could be bought with in game money got by just playing) - the more you have, the more you can try out and that's makes you a better player, but every week you can choose free champions, so this is not a big advantage. runes - which you can use it only in lvl 20 or higher, but if you are 20 lvl, you can buy them form IP got from just playing. And Riot Points which is good for buying another stuff in game.
On the other hand there is many stuff you can buy in game. But most of the stuff you buy in game (champions, runes, boosts) you can get by just playing and collecting IP. Except skins and themes which doesn't add any tactical advantage, only looks.