A poem that relates to Amir's struggle to find his identity is Dietrich Bonhoeffer's "Who am I?" which can be found at the following link: http://www.dbonhoeffer.org/who-was-db2.htm
Like Amir, Bonhoeffer was upper class. He, like Amir, returned to his country of origin, though, as Nazi Germany it was a dangerous place to be, after taking refuge for a time in the United States. Like Amir, he was haunted by his past, albeit in different ways from Amir.
Bonhoeffer ended up imprisoned by the Gestapo. In prison, because of his family and class, he was looked up to by the other prisoners, but like Amir, often felt that the successful front he put up was a fraud. The poem explores the gap between the person Bonhoeffer looked like on the outside and how he feels on the inside—an issue that Amir also grapples with.
Another poem that relates to The Kite Runner is "I Find No Peace" by Sir Thomas Wyatt. (See link below). The theme of the poem is loneliness and isolation, and the speaker likens his solitary condition to being shut away in a prison. The parallels with The Kite Runner are not hard to see. Amir is lonely and isolated, because he doesn't have the attention of his father, which he so desperately seeks.
There's an important line in the poem, where the speaker says:
I love another, and thus hate myself.
This could be used to refer to Amir's guilt over his betrayal of Hassan, which is closely related to the previous point. Amir craved his father's attention so much that, deep down, he knew that getting his father to send Hassan away would allow him to develop a closer relationship with Baba. In other words, Amir loves another—Baba—but hates himself over what he's done to Hassan.
This is definitely not a poem, but I do believe a great parallel text for The Kite Runner would be A Separate Peace. Compare Gene to Amir and the theme of envy.
You also might look at "Change" by Letitia Elizabeth Landon; it deals with two boys who are very close in their youth, but drift away as they get older and their priorities change.
"A Poison Tree" by William Blake is a great poem to relate to The Kite Runner. There is a link about the author below, followed by a link to the poem. It tells what happens when anger ruins a friendship. Anger is a kind of poison that can start small, like a seed, but if you feed it, it will grow into a tree. I think that's what happens to Amir in the novel, an anger and envy with Hassan that he feeds and feeds until it grows quite large. In the poem, the friend is killed by the anger, and of course, the actions of Amir, you could argue, based on his anger and envy, set the stage for Hassan to die. The poem and the book are powerful lessons about anger.