I am the character.
Their struggles are my own.
Their victories are my own.
Their failures are my own.
...well at least I'd like it to be most of the time.
As others have said, it depends on the game. It's sort of natural for me to project myself onto my PC's, but the game itself has to 'meet me halfway' so to say.
Deus Ex: Human Revolution, I felt like I was Adam.
Spec Ops: The Line, I felt like I was Walker.
The Walking Dead, I felt like I was Lee.
^And these are games with 'fixed' protagonists that we cannot alter. I still feel a strong connection with them all, enough to consider myself truly in their shoes.
Then there's several RPG's that let you create your own character. I also tend to see them as an extension of myself, even if I create some sort of otherworldly non-human character. It's odd though.
I never really saw 'myself' as Shepard in Mass Effect. No matter what, even though I was projecting myself onto him and would do things 'my way' if it really were me, I can't say I felt like I was Shepard. I always felt more like, I was controlling someone else.
Same thing with Dragon Age, I felt like I was one with my Dwarf commoner in Origins, and felt nothing of the sort for Hawke. (Why is it that I feel like I'm Adam, Walker, and Lee, and not Shepard or Hawke? Perhaps it's just something exclusive to Bioware.)
Then there's other games with characters I cannot comprehend forming a personal bond with. Such as every single person in Gears of War. No matter what, as much fun as the game gets, my body, heart, mind, and soul find it impossible to project onto anyone in the Gears series. I also can't with Kratos in GoW or Dante in DMC.
So I guess the game needs to do SOMETHING to draw me into it enough. Enough to consider myself truly in the shoes of another person. Some games do it, some just don't.