What do I get to putting my powers on my characters?

Am I correct in assuming you mean "What do I get for putting power on my characters?" At first it may not seem apparent, but outside of the core set, that are a lot of triggered effects that either require power, or give bonuses based on how much power is on the card. Devious Intentions comes to mind, you get +1 strength for every power on the character, plus stealth and an Intrigue icon. Another one for House Lannister is widows wail which lets you draw a card for each power after every challenge you win in which attached character participated. I may be a little off on either of those, as off this writing cardgamedb is down. Hope that gives you some idea.

It's a tradeoff. In addition to card effects (Ser Gregor's Band, for instance, gets a STR boost based on power you have that's not on your House; the new Jaime has a potent kill effect based on how much power he has on him; House Clegane Outlaw is cheap to play but has to have a power on him to be usable in challenges; and so on), you get put your power somewhere where it can't be taken by Power challenges (Brotherhood decks excepted, of course). But it's riskier because you lose all the power on a character if it's killed. So you have to decide how much risk you're willing to take each time you claim power.

Is your opponent beating you up with power challenges at the moment? You may want to use Infamy to make it harder for him to get at it if you can protect your characters from kills (dupes, claim sink, etc.). OTOH, if you're ahead on power but losing military challenges, you may want to put it on your House if you're short on claim sink.