The internal conflict in "The Cask of Amontillado" besets Montresor and may be described as the conflict of character against self, while the external conflict is a subtle one between Montresor and Fortunato and may be described as character against character.
Montresor's internal conflict has two aspects and has developed because he has received an unspecified "thousand injuries" and an insufferable "insult" from Fortunato, who seems completely unaware of (or blindly calloused to) all this injury and insult. One aspect of Montresor's internal conflict concerns bearing with the affront and insult until his "vowed revenge" might be enacted: "I had borne as I best could, but when he ventured upon insult I vowed revenge." The conflict here stems from feigning "good will" and deferring punishment.
The other aspect of his internal conflict concerns devising the perfect plan of revenge that might deflect any "idea of risk" away from himself: Montresor wanted a plan of such a nature that would ensure he wouldn't get caught: "I must not only punish but punish with impunity." The conflict here comprises thinking, thinking, thinking of the perfect way to punish him and protect himself. Both aspects of Montresor's internal conflict are nicely illustrated by this line:
I continued, as was my wont, to smile in his face, and he did not perceive that my smile now was at the thought of his immolation.
In this conflict, Montresor's internal focus was on bearing up and finding just the right plan to deliver at just the right time in just the right way so as to punish, to be the acknowledged avenger and to escape "retribution" for his deed.
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.
Montresor's external conflict, the subtle conflict between Montresor and Fortunato, was subtle because Montresor must keep up the subterfuge of "good will" so that he might win Fortunato's cooperation in his plan of vengeance. The plan necessitated luring Fortunato, under the guise of friendship, to a deep, dark place of entombment where Montresor could ridicule Fortunato (with the proffered Amontillado) and be rid of him in a manner that would properly assuage his desire for vengeance after having been wronged. That plan, the story tells us, was to wall Fortunato up alive, then walk away as his own "heart grew sick on account of the dampness of the catacombs." In one stroke, Montresor has thus conquered his foe and his external conflict.
I forced the last stone into its position; I plastered it up. Against the new masonry I re-erected the old rampart of bones.