In a story, the climax is that moment of greatest emotional intensity. In "The Cask of Amontillado," it comes when Montresor walls Fortunato up in a niche in the wall of the catacombs and Fortunato realizes what's happening to him. Prior to this moment, Fortunato didn't really know what was going on. He certainly had no idea of the gruesome fate that Montresor had in store for him. In addition, Fortunato was quite tipsy, which further impaired his judgement just when he needed it to be working properly.
Once Montresor has built the wall so that it's level with his chest, he shines a torch so he can see inside. At that moment, Fortunato lets out a series of blood-curdling screams which unnerve Montresor but not enough for him to free the hapless Fortunato from what will be his final resting place.
This is a moment of great emotional intensity as it's not entirely clear what Montresor will do at this point. Maybe he'll come to his senses and set Fortunato free, or perhaps he'll still exact revenge but in a less cruel and inhumane manner. But Montresor goes ahead with his wicked plan, and the resolution comes when he finished walling up Fortunato inside the niche, after which there are no more screams to be heard.