Most of the characterization done in this great short story is done indirectly. Readers are told some basics about what the sniper's face looks like in the beginning of the story.

His face was the face of a student, thin and ascetic, but his eyes had the cold gleam of the fanatic. They were deep and thoughtful, the eyes of a man who is used to looking at death.

This quote also seems to indicate that the sniper is an experienced killer; however, beyond those details, we are directly told little else. Readers are likely to figure out that the sniper is a creative thinker as well as somebody that can stay cool, calm, and collected during frantic, tense, and life threatening moments. We see this when the sniper refuses to panic despite his situation getting worse and worse. He is first spotted, then he is wounded, and finally he loses his main weapon. Despite all of those things putting him at a disadvantage, the sniper figures out a solution to his problem that allows him to kill his enemy with the remaining tools that he has on hand.

He is calm and patient.  The author makes this obvious by the sniper's ability to wait on his opponent and to remain almost motionless for a long period of time.

For most of the short story, the sniper appears to be detached and emotionless, but the ending causes the reader to question how such a collect character would proceed after discovering that he had killed his own brother.

The sniper is also a little bit of an Alpha male in that he seems to have extreme confidence in himself and "knows" that he will have victory over his opponent.