In this story, the Barber is secretly working as an informant for the rebels. Captain Torres, on the other hand, is tasked with bringing down the rebels and has already killed a great number of them. This explains why the Barber wants to kill him: it is an act of revenge for the murders that the Captain and his men have committed. It will also prevent the Captain from killing any more.
Later in the story, we learn of another reason why the Barber might want to kill Captain Torres, and it is expressed through the following lines:
And then on the other side. "The avenger of us all. A name to remember. (And here they would mention my name.) He was the town barber. No one knew he was defending our cause."
In other words, the Barber knows that if he kills Captain Torres, he will become a hero among the rebels. His prestige and reputation will be massively enhanced because he has killed the rebels' number one enemy.
Despite these motivating factors, the Barber is unable to kill the Captain. It seems that his desire to be a peaceful man and to simply be a barber outweighs his desire for revenge and prestige.