Three very easy choices for me start with John Martston from Red Dead Redemption. I won't spoil the ending, but I was definitely upset and easily chose the tale as the best videogame story I've seen. He is easily the most human character that I have played yet. My hat is off to Rockstar on this one, they win the contest.
That said, I can't help but a slightly greater connection to my Mass Effect FemShep. She is exactly what I would choose in her situation, and I am fully vested in her outcome and will rage out if she ends up dying in ME3. I probably never feel whole again lol (god I hope I am kidding).
HOWEVER, although what I said about both characters above is true, and I did not feel a closer connection to this character, Nariko from Heavenly Sword ultimately affected me the most. I'm going to say spoilers now, but it's a five year old game already. When Flying Fox drops Kai's body in a noose and Nariko falls to her knees screaming out Kai's name in tears, I definitely began to cry. I was so god damn pissed and wanted to gruesomely murder that mother f'er. I wanted to stomp on his face until there was nothing left... ahem. My point is that Heavenly Sword got the strongest reaction out of me, and it was because of the relationship between Nariko and Kai.