I'm happy to help you get started on that letter.  I can't write it for you though; Enotes frowns upon that kind of thing.  The story starts out with this line:

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

That tells you that Fortunato is likely a "user."  He uses his friends and acquaintances to his personal benefit.  It could also be that Montresor is completely ignorant of how his actions hurt people around him.  I doubt that Fortunato is actually physically injuring Montresor, so the injuries that Montresor is talking about are likely emotional injuries.  

Whatever Fortunato has been doing to Montresor is bad enough for Montresor to commit murder.  What would cause someone to seek murder?  I'm going to go with love.  I would focus your letter on how Fortunato knowingly or unknowingly stole the affections of a woman (or more than one woman) from Montresor to himself.  Perhaps one of Fortunato's sexual conquests was a family member of Montresor's.  The text doesn't indicate that Montresor has any family, but it also doesn't say that he doesn't have a family.  

The love angle could be your first set of injuries.  Another set of injuries could be business related.  Perhaps both men compete for the same kind of client base.  Or maybe they were business partners at one time, and Fortunato ended up somehow hurting their partnership or lost a lot of money.  

My last recommendation is to think about yourself.  What would make you so mad at a friend of yours to consider seriously hurting them back?  Write about that. 

What did Fortunato do to anger Montresor in "The Cask of Amontillado"?

This is, in essence, a creative writing assignment. Montresor never states explicitly why he is taking revenge on Fortunato. As you work on this assignment, you might try to focus on looking for subtle clues in the text, as that will give evidence that you read the story closely. There are two clues that might give you direction. The first is the description of the "damp ground of the catacombs of the Montresors" and the second is this segment of dialogue:

"The Montresors," I replied, "were a great and numerous family."
"I forget your arms."
"A huge human foot d'or, in a field azure; the foot crushes a serpent rampant ..."

What this suggests is two things: Montresor has a great deal of pride in belonging to an old, noble family, and some misfortune has befallen his family, as they were "great and numerous" in the past but are so no longer.

Thus the starting point for your letter might be trying to imagine something Fortunato did that caused financial or some other form of ruin to the Montresor family. Perhaps the Montresor family invested in a shipping company or wine import business partially owned by Fortunato that failed or Fortunato sold Montresor a shipment of wine that was of such low quality it could not be resold. Another possibility is that Montresor had hoped to marry a rich widow to repay family debts, but Fortunato managed to win the rich widow's affections, not realizing Montresor was a rival for her affections. 

The key element of the injury, though, is that Fortunato seems unaware he has harmed Montresor, so your choice of incident should be one in which Fortunato is not acting out of malice and remains ignorant of the hatred he inspired.