Traditional theater "science" viewed a drama performance as going through 5 acts. The first and second acts slowly built up tension until the 3rd act, when the major events (the climax) would occur, then the last two acts would wind down from there. This was graphed as a line which rose during the first half, then fell during the second, producing a mountain-like image.
This view has changed a bit in most modern dramas, as the show generally ends quickly after the climax. But the climax is still not an instant in time, but rather a closely-spaced sequence of events which somehow "resolve" all the tension that has built up to that point.
Thus, "at the climax" and "in the climax" convey two slightly different things. "At" refers to the situation at the very highest peak of drama, while "in" refers to actions during the scenes where the climax occurs. One is referring to the situation at a specific point in time and the other is referring to a period of time.