Montresor wants revenge against Fortunato for some imaginary insult.
It is important to remember that Montresor does not have a real reason for killing Fortunato. Montresor is a madman, and the reasons he wants to kill Fortunato are all imaginary. Poe makes sure that we realize this with his very first sentence.
The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge. You, who so well know the nature of my soul, will not suppose, however, that gave utterance to a threat. At length I would be avenged …
The hyperbole is our first clue. A thousand injuries? Really? Isn’t that a little extreme? You can’t really do a thousand terrible things to a person, especially without him noticing. Montresor probably imagines these, or turns minor slights and insults into crimes worthy of prosecution.
Montresor wants to not get caught. He explains that he has to get away with the murder, or he will not really be avenged. Since the telling of the story is fifty years after the event, he must have succeeded.
Another reason that we know that Fortunato did not really do something terrible enough to Montresor to justify being killed is that he agrees to go with Montresor into the catacombs. If you had wronged a person in some terrible way, you would not go underground with him at night without witnesses.
Montresor is able to easily convince Fortunato to go into the catacombs with him by telling him he has a cask of valuable Amontillado wine that he needs his opinion of. When he offers to show it to Luchesi instead, Fortunato protests.
“Come, let us go."
"To your vaults."
"My friend, no; I will not impose upon your good nature. I perceive you have an engagement. Luchresi--"
"I have no engagement; --come."
Montresor makes other arguments that he should not go, such as the fact that it will not be good for his cold, but Fortunato insists. He willingly goes underground with Montresor because he does not know that the man is his enemy.
If Fortunato had really done something to Montresor, it would be a very different story. This one is the tale of a crazy guy killing another guy for no reason. He is a psychopath, because he believes that he is right. Montresor really thinks that Fortunato deserves to die and has no problem with killing him as long as he gets away with it.