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.