This is a very astute question to ask about this incredibly Gothic text of Poe's. Poe is definitely a master of creating narrators who are unreliable, whose account and version of affairs we begin to question as we progress through the story. The first paragraph makes it clear the way that Montresor considers that he has been insulted and how he feels a compulsion to gain revenge:

The thousand injuries of Fortunato I had borne as best I could; but when he ventured upon insult, I vowed revenge.

As we read the rest of the story, however, we begin to question the veracity of this statement. Such a character as Montresor, who coldly and calculatingly plans and carries out a terrible premeditated murder, does not appear to be able to bear any number of "injuries" very well, let alone a thousand. Also, the way that Fortunato greets Montresor and is happy to go with him into the catacombs without fear for his own safety does beg the question of to what extent this "insult" is actually real or not.

Apart from this, the way that Montresor jokes and plays with Fortunato whilst he is bricking him in is really disturbing, and certainly Montresor is a narrator that appears to be very unaware of his own feelings and emotions. Note how in the last paragraph he admits his own feelings of sickness at what has happened only to go on and explain them away by some other means:

My heart grew sick--on account of the dampness of the catacombs.

Montresor, in conclusion, appears to be a very disturbed individual who you wouldn't want for a friend! He is clearly psychologically unstable and what is more worrying, able to act on his various neuroses and sense of paranoia in a highly intelligent and sophisticated manner to gain his revenge.

The narrator of "The Cask of Amontillado" is vindictive, spiteful, quick to anger, two-faced, extremely clever and meticulous, and deranged.  In short, he is a megalomaniac: focused entirely upon the act of revenge instead of the original reason which drove him to it.

Here are some quotes that reveal his characteristics:

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 definitely, settled—but the very definitiveness with which it was resolved precluded the idea of risk. I must not only punish but punish with impunity.

Even though the narrator uses wonderful word choice and a kind of ethical argument to describe revenge, he gives no motive or justification for what Fortunato has done to drive him to revenge.  Could he be making it all up: both the lack of cause and the act itself?

A succession of loud and shrill screams, bursting suddenly from the throat of the chained form, seemed to thrust me violently back...I replied to the yells of him who clamoured. I re-echoed, I aided, I surpassed them in volume and in strength. I did this, and the clamourer grew still.

The narrator's yelling even louder than a soon-to-be dead man shows that he emotionally revels in his sadistic act.  At a time when he should show a sign of guilt, Montressor reveals none: only shouts of joy.

For the half of a century no mortal has disturbed them. In pace requiescat! ("May he rest in peace!")

The narrator has told use--in sparkling detail--every part of his plan and execution of revenge, and--come to find out--he is an old man, at least 70 years old.  Is he not a deranged old man?  Or, is he making up the story completely?  Which is the more crazy?

Edgar Allan Poe, who wrote "The Cask of Amontillado," did an excellent job of conveying his narrator's unstable state of mind throughout the story.  Poe's used first person narrative very cleverly and skillfully in order to lead the reader to create a clear mental impression of being told a story by a somewhat derange individual (Montresor, the narrator) who is obsessed with revenge, as well as being fully aware of his own cunning and success as a murderer.

The reader begins to perceive that Montresor is not quite "right" when he details his prerequisites for the act of revenge, which are found within the first paragraph.  He quickly adds that his "smile now was at the thought of his immolation," which is not a particularly sane view of his nemesis.  Montresor's pretense of concern for Fortunato's health, as well as his apparent pleasure at the naivete of his victim, are indicators of his lack of mental soundness.

Although Montresor seems to do a reasonably good job of maintaining a semblance of mental normalcy, there is one point in the story in which he appears to give in to his mind's illness:

...For a brief moment I hesitated, I trembled.  Unsheathing my rapier, I began to grope with it about the recess; but the thought of an instant reassured me.  I placed my hand upon the solid fabric of the catacombs, and felt satisfied.  I reapproached the wall; I replied to the yells of him who clamored.  I reechoed, I aided, I surpassed them in volume and in strength.  I did this, and the clamorer grew still.

Although Montresor regains composure after his outburst, the reader is left with no doubt about his lack of mental stability.