Explain the crime the narrator had committed and how he was found out?

I think saying that the narrator committed more than one crime is more appropriate. If a country doesn't have a legal law about torturing animals and abusing a spouse, then those are still moral crimes. With that said, I do believe that the question is asking about the narrator’s final crime and cover-up. While walking downstairs, the narrator is tripped up by his replacement cat. Enraged, he tries to kill the cat with an ax. His wife stops him from doing that. Unfortunately, his anger turns to her, and the narrator kills his wife. He then decides to hide her body in the cellar wall.

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.

Unbeknownst to the narrator, the cat gets walled up too. Four days later, the police come and search the home, but they find nothing. As they are making to leave, the narrator taps on the wall and comments about how sturdy the construction is. The noise causes the cat to begin wailing in response. The police tear apart the wall, discover the cat and the wife's body, and arrest the narrator.

Upon its head, with red extended mouth and solitary eye of fire, sat the hideous beast whose craft had seduced me into murder, and whose informing voice had consigned me to the hangman. I had walled the monster up within the tomb.