This is a masterful tale of suspense and tension, and so you will want to focus on how the character description builds tension and reveals the fear and terror that the barber is undergoing. But let us start with essentials. We know two essential qualities about the barber, as he himself tells us:
Yes, I was secretly a rebel, but I was also a conscientious barber, and proud of the precision required of my profession.
It is these two qualities that really shape the barber as a character. As he faces the dilemma of whether or not to kill Captain Torres, he comes back to the fact that he is a barber who takes pride in his work:
I'm a barber who does things properly...
There, surely, the razor had to be handled masterfully, since the hair, although softer, grew into little swirls. A curly beard. One of the tiny pores could open up and issue forth its pearl of blood, but a good barber prides himself on never allowing this to happen to a customer.
In spite of the chance that he had, what is more important to him is his profession and the fact that as he himself says, he is "the best [barber] in the town."
However, this opportunity also allows the barber to show his moral values and his true superiority over Captain Torres. As the story progresses it moves to an almost stream-of-consciousness kind of narration, where we hear the internal monologue of the barber's thoughts. Consider the following passage:
Damn him for coming! I'm a revolutionary, not a murderer. And how easy it would be to kill him. And he deserves it. Does he? No! What the devil! No one deserves to have someone else make the sacrifice of becoming a murderer. What do you gain by it? Nothing. Others come along and still others, and the first ones kill the second one, and they the next ones - and it goes on like this until everything is a sea of blood.
This, to me, is a very revealing passage, because it shows that the barber has higher ethical standards than the Captain. The barber recognises the futility of murder, and even though he has been given a golden opportunity to help the rebel cause, if he takes advantage of it he will be lessening his own humanity by becoming a murderer, because in the end, murder achieves nothing.
Thus it is these two characteristics - the pride that he takes in his work and his moral outlook on life - that essentially define the barber and ensure that he makes the right choice. This is a right choice that Torres himself acknowledges, when he leaves the barbershop saying "But killing isn't easy." Is this perhaps a grudging mark of respect for the stand that the barber has taken?