Let's start by defining existentialism. The Routledge Encyclopedia of Philosophy highlights some of the key ideas that we can then apply to The Kite Runner. In the article on existentialism in Routledge, Charles B. Guignon asserts that

First, existentialists hold that humans have no pregiven purpose or essence laid out for them by God or by nature; it is up to each one of us to decide who and what we are through our own actions. . . . Second, existentialists hold that people decide their own fates and are responsible for what they make of their lives. . . . Finally, existentialists are concerned with identifying the most authentic and fulfilling way of life possible for individuals.

The connecting thread here is that existentialism emphasizes human free will. As such, we can say that Amir in The Kite Runner has free will to make choices that affect his life. Therefore, while the choice to abandon Hassan in the alley during his assault falls on Amir's shoulders, the decision to try to redeem himself is also available to him.

Amir's childhood is marked by both his friendship with Hassan, who is also his servant and cultural "inferiority" according to Afghani society, and his strained relationship with his father, who thinks Amir is not "manly" enough because he doesn't stand up for himself and wants to be a writer. Amir struggles with his identity as a child, hoping to win his father's love but usually failing to gain his approval. There is a mismatch between who he feels he is inside and who his father expects him to be. Amir sees a chance to resolve this conflict with the kite-fighting tournament; the interest in kite-fighting is one commonality between Amir and Baba, and Amir believes he can win his father's love by winning the tournament and bringing home the second place kite. Hassan is the kite runner, and he does indeed run down the second place kite when Amir cuts it down to win the contest; however, Assef and his friends find Hassan and assault him in the alley. Amir sees what happens but runs away, so his victory is always tainted by his failure to help Hassan. Amir gets what he wants—his father is very proud of him—but it comes at a very high cost. Amir's guilt haunts him, causing him to frame Hassan for theft to get Hassan and his father removed from the household (they leave because Ali is indignant that the pure-hearted Hassan has been accused and also because Amir has been mistreating Hassan). Amir and Baba later start over in America due to the political upheaval in Afghanistan; Amir is hopeful that he can forget about Hassan and his guilt there.

Amir finds, though, that he cannot outrun his problems. His lack of action can only be later alleviated by a trip back to his homeland when his father's best friend Rahim Khan (and pseudo father figure to Amir) calls Amir when he is an adult and tells him "there is a way to be good again." Amir then must choose to travel to the Middle East and save Hassan's son Sohrab from Assef. He must make the choices along the way to defend Sohrab, and eventually to fight (both physically and legally) to adopt Sohrab and bring him to California. All of the decisions Amir makes along the way allow him to at least begin to absolve himself of the guilt he has felt for so many years. He uses his free will to take up this redemption journey. Along the way, he discovers secrets about his family and reevaluates his relationship with his father and his sense of self.

The fact that Amir has a chance, a choice, offered to him in the opening of the novel is where one can see the presence of Existentialism in the novel.  Granted, that this choice is to "become good again."  Yet, this is the choice that is in front of him.  It is choice and the consequences from this that ends up playing a formative role in Amir's sense of self. Nothing is absolute.  Nothing is transcendent.  In true Existentialist fashion, Amir must provide the justification for his being in the world.  His desire to return to Afghanistan, to right that which is wrong, to care for Sohrab, and to be "the kite runner" are all examples of the Existentialist idea that individual freedom and action ends up defining one's identity.  Identity is not something that is transcendent.  In the best example of "existence precedes essence," Amir ends up defining his identity through his choice. There is only this choice that defines who he is and what he sees his identity to be.  Amir is not shown as one who depends on the transcendent power of the divine to define his identity.  It is Amir's own power of choice that defines his identity.  This becomes an essential element of Existentialism for it places priority on the individual for defining their own identity.  In this case, Amir becomes a protagonist that can be seen in Existentialist in defining his own identity.