The first paragraph of Edgar Allan Poe's "The Cask of Amontillado" explains exactly why the narrator of the story (we learn later that it is Montresor) is so intent on killing Fortunato. It is not a particularly detailed explanation, and we are immediately struck by how little information Montresor actually gives us.
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 I gave utterance to a threat. AT LENGTH I would be avenged; this was a point definitively settled -- but the very definitiveness with which it was resolved precluded the idea of risk. I must not only punish, but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.
Looking at this explanation closely does not help much, but we do learn a few things. First, the narrator believes he has been "injured" by Fortunato at least a thousand times. Secondly, none of these so-called injuries has ever risen to the level of an "insult" until now. That is it. No more. The rest of the paragraph indicates that Montresor intends to get revenge. The best revenge, he says, can only happen if Fortunato feels as "injured" as Montresor feels and if Montresor is never caught or punished for the revenge he takes.
As the story unfolds, it is interesting to note that Fortunato does not appear to have any kind of bad feelings about Montresor, nor does he seem to suspect that Montresor has any kind of nefarious plans against him. It is interesting to consider, then, whether Montresor really accomplishes his goal of "mak[ing] himself felt" by Fortunato, even at the end of the story.
One more thing to consider is whether or not Montresor is a reliable narrator. In other words, are the things he says true or are they the rather delusional beliefs of a rather deranged man? If the latter is true, Fortunato may have done absolutely nothing to deserve the fate Montresor delivers him.
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.