The narrator is telling the story because of the terrible burden of guilt he feels over his crimes. One of these is of course the murder of his wife but he feels perhaps just as guilty about his treatment of Pluto, his most beloved pet. He recounts that these events "have tortured" him and "have destroyed" him.
He recounts in detail the great fondness he had for Pluto. This cat, amongst all his other animals, was the most dear to him. The cat demonstrated an obvious affection for him and would have followed him everywhere if possible. But once the narrator succumbs to the demon of alcoholism, he quickly descends into a dark place where he begins to feel a terrible annoyance and anger about Pluto. The cat had begun to avoid him as he was in a terrible state nearly constantly.
Once he cuts out Pluto's eye in a fit of rage, he decides that he must hang the cat. This is not to put the cat out of its misery but in fact is to damn himself. The narrator feels so guilty that he feels the need to place his soul beyond any hope of mercy.