The point of view in "The Cask of Amontillado" is first-person, also called first-person narrative; this style uses "I" or "We." In first person point of view, the text is narrated by a character in the story. In this story, Montresor is the narrator and main character. We (readers) get the story from Montresor's point of view.
Using the first person point of view, a writer can choose to give the reader a particular perspective. Montresor does not reveal every detail so this makes him an unreliable narrator. Montresor is vague about why he is so angry with Fortunato. He claims that he has suffered a thousand injuries from Fortunato with the last straw being an "insult." The reader has no other information about how Fortunato has wronged Montresor. Montresor insists that his revenge must be drastic, but this doesn't clue the reader in on whether or not Fortunato deserves his fate.
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.
It would be logical to assume that Fortunato wronged Montresor in a terrible way considering how angry Montresor is. However, if Montresor is insane, unreasonable, or overly sensitive, it would be logical to assume that Montersor is just overreacting and/or behaving like a psychopath. This is the effect of using an unreliable, first person narrator; the reader simply can not be sure about Montresor's justifications. Had the story been written with a third person, omniscient narrator, the reader would have a more objective perspective most likely with Fortunato's side of the story.
Edgar Allan Poe had a choice to make, as all authors do, when he wrote "The Cask of Amontillado." He had to decide who was going to tell the story. In this case he had three choices: Montresor, Fortunato, or a disconnected narrator.
Consider what the story would have been if Fortunato had told the story. It is tempting to think that we may have learned something more about the supposed insults he offered Montresor which apparently prompted this entire episode; however, it seems unlikely that Fortunato actually did or said anything insulting, so all we would have gotten was a story full of unanswered questions and confusion. In fact, even as Montresor was putting the last brick in the wall, it is not clear that Fortunato really understood what was happening to him or why. He would not have been an effective teller of this story.
A detached narrator might have worked, would certainly have worked better than Fortunato; however, we certainly would not experience quite the same horror as when we hear Montresor talk about his unholy acts. We might have had the facts but not experienced the same reaction.
So, Poe chose to use the first-person point of view so Montresor could tell his own story and attempt to justify his unjustifiable actions. We hear Montresor's voice when he tries to justify his horrific plan:
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.
It is obviously told in the first person because the narrator uses first-person pronouns and shares with us his thoughts. Unfortunately for us (the readers), Montresor is not a reliable narrator. We know that a man capable of such a cold-blooded act cannot be completely sane, and we see no evidence that Fortunato has done any of the things Montresor accuses him of doing. Despite that, we are privy to what Montresor is thinking as he commits this heinous act because he is the narrator, and this makes Poe's choice of Montresor as narrator the perfect choice for this horror story.