The climax of a story is often the most exiting point in the action, but since the entire story of "The Most Dangerous Game" is pretty exciting, locating the climax isn't a simple as it may be with other works of fiction.
For this story, you need to examine the events carefully to determine the main turning point, which is another indicator of the climax. Where in the story does the action or a character change?
I think the most obvious answer is near the end of the story when Rainsford confronts Zaroff in his bedroom. Zaroff tells Rainsford that he is free to go, since he won the game. At this point Rainsford has the opportunity to show his moral superiority to Zaroff, but instead, he chooses to continue the game. Interestingly enough, after having killed Zaroff, Rainsford does NOT free the "prey" that Zarroff has been holding for future hunts. This inaction further supports the idea that Rainsford's confrontation with Zaroff is the climax of the story, since the possibility of his replacing Zaroff (symbolized by a comfortable night's sleep in Zaroff's bed) definitely indicates a change, or turning point, that takes place in Rainsford's character.