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!