To properly answer the question of what made the narrator kill his wife in the short story "The Black Cat" by Edgar Allan Poe, it is important to briefly go over the sequence of events and the author's clues about the narrator's character.
The narrator explains that since he was young he loved animals and had a large number of pets. When he marries, his wife also loves pets, and so they have several. His favorite is a black cat named Pluto, whom he treats especially well. However, the narrator succumbs to "the fiend intemperance." In other words, he starts drinking alcohol heavily, and as the alcohol begins to affect his mind and judgment, he starts abusing the animals. He refrains from hurting Pluto until the cat slightly bites him in fright. In rage, the narrator first puts out one of the cat's eyes with a pen knife and then later hangs it by a rope from the branch of a tree.
That night the house catches fire, and a silhouette of a large cat appears on a wall. This image haunts the narrator, and he decides to find another cat to replace Pluto. The new cat follows the narrator everywhere, but the narrator's guilt and mental instability causes him to become disgusted, annoyed, and terrified by it. One day, the narrator and his wife are descending the steps into the cellar, and the cat accompanies them. His instability due to his fear of the cat causes the narrator to pick up an axe and attempt to kill it. However, his wife stops him from delivering the fatal blow. The narrator, experiencing "a rage more than demoniacal," turns the axe on his wife, burying it in her brain and killing her.
We see then that a long period of mental instability reaches its peak in the narrator's hatred of the second black cat. When his wife tries to keep him from killing it, the narrator turns his rage on his wife and kills her with the axe.