Often they're a character that isn't me, but that I control and try to stay true to. That said it really depends how linear the game is. I mean, CoD vs Skyrim vs Deus Ex Human Revolution, all different levels of versatility. CoD just shafts you through and it doesn't matter who you are, Skyrim lets you do practically anything and doesn't make any effort to impose a character onto you, and DE:HR gives you a character that has their own motivations and thoughts but still allows you to choose their general ideology and gameplay style.
That said though, I'm hopeless at staying true to character if there's stuff to be had. My virtuous warrior often ends up as a mostly virtuous warrior who will stoop to any low to get a new weapon, and my ruthless assassin often ends up as a ruthless assassin who still has time to rescue your cat or collect 10 tiddlywinks even though it's a complete waste of his time.
That said though, I'm hopeless at staying true to character if there's stuff to be had. My virtuous warrior often ends up as a mostly virtuous warrior who will stoop to any low to get a new weapon, and my ruthless assassin often ends up as a ruthless assassin who still has time to rescue your cat or collect 10 tiddlywinks even though it's a complete waste of his time.