I think one stylistic element you will want to focus on in the climax that really helps to make it incredibly dramatic is the use of onomatopoiea that seems to capture the madness of the narrator. The description of the sound of the heart and its impact on the narrator and how he responds serves to really highlight the frenzied state of the climax.
You may want to write an essay that justifies this climax. (i.e. the climax is appropriate/fits the plot/story). That is, after you identify the climax, you want to explain how all the events of the exposition and rising action build to this climax. Then, show how the falling action is the logical result of this climax. (The climax is your pivotal focus while you explain how the action leads to it and follows it logically.)
As you know, Poe is credited with the creation of the modern short story form. He felt that a story should have a single focus. "Tell-Tale Heart" is a great example in the way in which Poe builds the tone of horror to the climax. For instance, in the exposition (beginning), Poe's narrator immediately appears mentally deranged: "True--Nervous--very nervous, dreadfully nervous I had been and am....I loved the old man...He had never given me insult.For his gold I had no desire. I think it was his eye!...." The narrator's use of exclamation points and obsession with the eye--but he loves the man?--indicates mental abnormality. It is this initial nervous pitch and obsession that carry the action to the climax, is it not?
See the site below to give you some background on Poe and his techniques which you may want to incorporate into the introduction of your essay. Your thesis can be on the techniques that Poe uses to make his climax appropriate to the story.