How does Edmund's character evolve in The Man Who Was Poe?

Edmund becomes more and more assertive throughout the book. In the beginning of the book, he is very timid and compliant. Edmund and his sister have been left in their room by their Aunty, who has gone away and not returned for two days. There is nothing left to eat in the house, and Edmund's sister, Sis, believes they need to take charge and look out for themselves. Edmund, though, is hesitant. Aunty has instructed them to stay put, no matter what, and he does not think they should disobey, under any circumstances. Edmund has great respect for authority and is used to doing "what he is told" by adults, to the point that he, unlike Sis, is afraid to think for himself.

In the middle of the story, Edmund, bewildered by Dupin's erratic behavior, begins to question whether adults are, indeed, always correct and worthy of unqualified obedience. Although he is still very compliant to directions given him by adults, he acts on his doubts on occasion, interrupting Dupin in mostly futile attempts to keep him on track. Edmund would never have imposed his own will on an adult previously, but his experience with Dupin and his desperation to find Sis leads him to assert himself at times.

By the end of the story, Edmund, although still timid, has enough gumption to openly defy Dupin, who, despite the fact that he is an adult, is clearly not in full possession of his faculties. He manipulates his conversations with the man in order to keep him active in the search for Sis, and even snatches Dupin's precious notebook and threatens to destroy it if Dupin does not focus on the task at hand and find his sister.

At the beginning of the story in Avi's The Man Who Was Poe, Edmund trusted more in adults' abilities to help him, a mere child, than he trusted in his own abilities. Though, at the start, he wasn't sure Edgar Allan Poe, calling himself Mr. Dupin, would truly help him, Edmund believed he needed to rely on the help of some adult, so he chose Mr. Dupin. By the end of the novel, however, Edmund is really the main person who rescues his sister, not the adults around him. Edmund's successful rescue of his sister helps him realize his ability to accomplish his goals, without the help of adults. In addition, Edmund comes to realize that the goals of adults, like the goals of Poe, may not fully align with Edmund's own goals, making any help adults can provide very limited.

Edmund realizes the limited abilities of the help Poe can offer the moment Edmund comes to understand Poe wasn't truly interested in saving Sis's life; rather, he was only interested in writing an interesting story. We see Edmund reach this realization when he suddenly asks Poe the following in the final chapter:

You never did want to save my sister, did you? ... You only wanted to make sure she'd die (Chapter 22).

One reason why Edmund's assessment of Poe's motives is true is because Poe saw such a strong parallel between Edmund's life and his own life. In Poe's own life, Poe's wife, named Victoria but called Sis, died very young. Unable to escape the torment of her loss, Poe can't help but see the death of a Sis, or even all death, as inevitable. In Poe's mind, the only way to make someone live forever is by capturing the person's story on paper, which is what he means when he gives the following reply to Edmund's comment:

I ask you: in what fashion will your sister live longer? In her life? Or, in this, my story that would have been? (Chapter 22)

What was just a story to Poe was Edmund's real life, and unlike Poe, Edmund had a chance to guide the ending of his own story by rescuing his own Sis. To do that, he had to separate himself from Poe and Poe's own goals, which is what Edmund came to realize by the end of the story.

How does Edmund's dialogue help portray his character in The Man Who Was Poe?

Edmund is a shy and mindful young man in the beginning.  His dialogue with his sister is all about what they should and should not do, according to Auntie.  Edmund also is painted to be very inquisitive.  He is constantly asking questions and begging for answers.  By the end of the story Edmund becomes more vocal, more outspoken and more independent.  All of these characteristics are demonstrated by Edmund and "painted" by Avi through the dialogue Edmund has with characters throughout the novel.  Edmund tells Sis that they can't leave the apartment because they promised their aunt that they would not leave until she returned. He also questions almost every statement made by Auguste Dupin.  He remembers that his Aunt taught him that "adults are always right, because they are the adults."