The sniper killed a woman in cold blood. What does this say about his character in "The Sniper"?
I'm not sure I would be so quick to characterize the sniper's killing of the woman as a cold-blooded act. He is a soldier engaged in war and the old woman has (likely) just revealed his location to the soldier in the car. The sniper does not kill just the old woman; he kills both her and the soldier she has informed. His act is one of survival and self-preservation. To not act would have likely meant his own capture or death.
Certainly, the killing of an old woman is something we would consider as being cold-blooded in many situations. However, this case may be an exception. Immediately following their deaths we also see some shades of guilt and remorse on the part of the sniper as well. He has mixed feelings about what he has just done. War can put individuals in horrible situations where such decisions are forced upon them. Horrific, regrettable, life-altering are all phrases that describe the sniper's action; cold-blooded may not be the best fit.
I think that the most obvious answer to this is that this action shows that the sniper is a ruthless man. He is willing to shoot even an old woman. To most of us, this would seem like an incomprehensible act because we are supposed to treat the old more gently than others. So his killing an old woman shows that he will do anything for his cause.
I think you can also say that he is a vengeful person. When he shoots her, she has already told the soldiers where he is. Killing her will not take that knowledge away from the soldiers. So his killing her shows that he wants to take revenge on people who try to hurt him.
So, overall, I'd say this shows him to be a ruthless and vengeful person at this point in the story.