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.
The direct answer to your question is that our unreliable narrator kills his wife because she tries to prevent him from killing the cat, which is a "replacement" for Pluto, another cat that the narrator has already disfigured and subsequently murdered. If we look a little deeper, however, I would argue that the narrator killed his wife due to his ongoing mental instability and addiction to alcohol.
To the narrator's way of thinking, it is the cat's fault that he murdered his wife. To put it mildly, mental illness and alcoholism have taken their toll on this man, and ironically, it is the guilt he feels over the murder of the first cat, Pluto, that leads to the murder of the second cat. It is thanks to his wife's effort to spare the second cat's life that he sees red and kills her.
On the day in question, the narrator and his wife go down to the cellar of the house that they...
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are living in since their original house burned down. Pluto's replacement follows them down the stairs, and this leads the narrator to another fit of irrational rage. He raises an axe with which he plans to kill the cat, and his wife steps in to stop him. Instead of the cat, it is his wife's brain into which he buries the axe.
Further Reading
The narrator of seems to blame the root of all his problems in "The Black Cat" on alcohol despite the fact that alcohol doesn't make most people suddenly abuse their loved ones. This inconsistency aside, it's drunkenness that the narrator says makes him abuse his animals until one day the abuse goes so far that he gouges out his cat's eye and kills it.
Soon after, another black cat appears who also, strangely, is missing an eye, and he adopts it to replace the one that he murdered. It's this cat who the narrator is too afraid to abuse (and who torments him) that he blames for his increasingly abusive behavior toward his wife. One day he tries to kill the cat (again?), but his wife defends it, and he kills her instead.
The narrator effectively blames the cat for his killing of his wife. It was the cat—the second cat—that reminded him of his first cat, Pluto, the one he got mad at and hanged. He was just about to hand his second cat a one-way ticket to kitty heaven when his wife interrupted him. This was the last thing she ever did because her crazed husband then turned on her with an ax, and that was that. If the cat hadn't been such a bad kitty, if it hadn't reminded the narrator of his previous bad kitty, then he wouldn't gone completely berserk.
At least that's what the narrator would have us believe. But he, like so many of Poe's narrators, is completely unreliable. That being the case, it's probably best if we just put the narrator's wicked actions down to a toxic mixture of mental illness and booze. The precise nature of their influence, however, is largely a matter for trained psychologists.
Your original question unfortunately contained more than one question, and so according to enotes regulations I have had to edit it down to one question, focussing on why the narrator killed his wife.
Crucial to understanding this story is understanding the influence of the black cat on the narrator and how what he does to it and then the continual haunting of the black cat drives him ever madder and madder. The guilt he feels at having killed the first black cat, and then the arrival of the second cat that always haunts him drives him over the edge into insanity:
Beneath the pressure of torments such as these, the feeble remnant of the good within me succumbed. Evil thoughts became my sole intimates - the darkest and most evil of thoughts. The moodiness of my usual temper increased to hatred of all things and of all mankind; while, from the sudden, frequent and ungovernable outbursts of a fury to which I now blindly abandoned myself, my uncomplaining wife, alas! was the most usual and the most patient of sufferers.
It is in the immediate paragraph following this that the narrator and his wife go down to the cellar. So angered by the presence of the black cat ever following him, the narrator tried to "aim a blow" at the cat with an axe. His wife stops him, and, in a moment of sheer madness and anger, he brains her.
Thus the narrator kills his wife because of the madness into which he has been driven thanks to the black cat and its continual, haunting presence around him, never giving him a moments peace.
Why does the narrator in "The Black Cat" kill his wife?
The narrator of "The Black Cat" kills his wife because she dares to prevent him from killing their pet cat. As he goes down into their cellar one day, she goes with him, and the cat nearly trips him on the steps. In the past, his fearfulness of the animal prevented him from harming it, but this sent him into a rage. Holding an axe, he lifts it and aims a blow at the cat in order to kill it, but his wife stays his hand. Already "exasperated ... to madness" by the fact that the cat nearly tripped him and sent him "headlong" to the bottom of the stairs, he then swings the axe down onto his wife's head, burying it "in her brain." She falls dead immediately.
All she did, then, was attempt to prevent the narrator from killing their cat with an axe—which seems a perfectly reasonable thing to try to keep someone from doing—and so he murdered his wife instead. After the fact, he does not even express remorse for what he did. He quickly figures out what to do with her body so that his guilt will not be detected, and he makes plans to wall her corpse up behind some bricks and plaster in their basement. He deceives the police officers who come to investigate his wife's disappearance, and he very nearly gets away with his crime, which is revealed by the black cat.
In "The Black Cat," how did the husband kill the wife?
I had to edit your question down to one question according to enotes regulations. The brief answer to your question is that I would encourage you to read this excellent story - it is not long at all and is an excellent example of the craft and art of Poe in all of his gothic and psychological terror. However, to respond to your first question, the narrator's murder of his wife is clearly linked to the presence of the black cat, that drives the narrator into a murderous rage:
One day she accompanied me, upon some household errand, into the cellar of the old building which our poverty compelled us to inhabit. The cat followed me down the steep stairs, and, nearly throwing me headlong, exasperated me to madness. Uplifting an axe, and forgetting, in my wrath, the childish dread which had hitherto stayed my hand, I aimed a blow at the animal which, of course, would have proved instantly fatal had it descended as I had wished. But this blow was arrested by the hand of my wife. Goaded, by the interference, into a rage more than demoniacal, I withdrew my arm from her grasp and buried the axe in her brain. She fell dead upon the spot, without a groan.
So, in trying to kill and dispose of the continual haunting presence of the black cat, the narrator is stopped by his wife. Instinctively he is driven into a murderous anger and without thinking turns the axe on her, burying it deep into her head, and thus killing her.
Now, read the rest of the story to find the answer to your second question which I was unable to answer about where he hid the body!