For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
We're in luck. "The Purloined Letter" is neatly divided into three acts. In that respect it follows a classic cinematic structure—or, considering that it was published well before "cinematic" became an adjective, maybe just a classic dramatic structure.
In this first act, we meet our three characters and get the background info to the major mystery: Dupin, the narrator, and G— discuss (1) the titular purloining; and (2) the way G— has been searching D—'s suite (and half the neighborhood around it) every night for the past three months. This act ends when G— leaves the smoky library, looking "entirely depressed in spirits" (73).
Act II
A month later, the narrator and Dupin are in the library, smoking. Hm, this sounds familiar. G— once more shows up. (Boooring.) He still hasn't found the letter. (Yawn.)
Dupin steers the conversation to the topic of the humongous reward being offered for the letter. G— says he'd give a pretty penny of his personal money for that letter. Dupin tells G— to write a check for the amount. After he does, Dupin gives him the letter. Whoa! The act ends with G— walking off with the letter. He's flabbergasted. So are we.
Act III
After G— leaves, Dupin explains his theory of detection. This takes a while and includes some childhood anecdotes. When he finally finishes, he tells the narrator that this theory helped him find the letter, which was in plain sight at D—'s place. Finally, he reveals that he was motivated by more than just the money. He sought out the letter out of loyalty (political and perhaps personal) to the royal lady, and because he really wanted to get even with D— for a past insult. Dun dun dun.