We never learn what, specifically, the "thousand injuries" Montresor refers to initially are. However, we do see Fortunato insult Montresor in the story itself, in more or less obvious ways.
When Montresor first happens upon Fortunato, he explains that he has purchased a large quantity of Amontillado, a Spanish wine, and he expresses his doubts that it is actually Amontillado. Rather than say, for example, "Oh, dear, my friend! Let's figure it out together," or something of this nature, Fortunato says,
How?" [...]. Amontillado? A pipe? Impossible! And in the middle of the carnival!
Fortunato is immediately dismissive. Montresor could likely see this as an insult to himself, as though his expertise and connoisseurship of wine is so negligible to Fortunato as to be completely discounted. Fortunato states, further, "You have been imposed upon," as if utterly dismissing the doubts Montresor has already expressed, further implying that Montresor has been duped.
As the men walk through the vaults, Montresor expresses concern for Fortunato's health. His guest keeps coughing and stumbling and is clearly not very well, but Fortunato again dismisses Montresor, saying,
Enough, [...] the cough is a mere nothing; it will not kill me. I shall not die of a cough.
This is rather rude of Fortunato, considering how Montresor must appear: as a concerned friend who wants to protect him. Fortunato speaks to Montresor as though Fortunato believes himself to be superior—more knowledgeable, more respectable, and so on—and this could be considered insulting.
Later, Fortunato makes an effort to identity Montresor as one who is "not of the brotherhood" of the masons. This seems to be yet another way in which he hopes to insult Montresor and to prove his own importance and superiority.