Not much direct information is given about the character of the Republican sniper in the short story "The Sniper". The reader is told that the sniper had "the cold gleam of the fanatic". A fanatic is a person who supports a cause with a lot of enthusiasm. A person who will do whatever it takes to support that cause, even if it's illegal or is dangerous to risk life and limb. Therefore, by telling us he has the look of a fanatic, we can infer that he will do what it takes to fight for the cause of his army. The reader can also see that the sniper must be young based on the following quote from the story, "His face was the face of a student, thin and ascetic..."
That's really about it for direct characterization. The rest of the characterization is given to us indirectly based on the sniper's actions, and some physical characteristics. We can infer that he is an experienced soldier. In the story, we are told that his eyes were "deep and thoughtful, the eyes of a man used to looking at death." We can infer that this isn't his first stint in combat. He has fought numerous times prior and has probably killed many times or has seen many killed. Also, he kills the sniper in the turret with one shot and the female informer in one shot. His aim is impeccable. That is from years of experience.