How must Amir contend with some aspect of the past? And how does his relationship with the past contribute to the overall meaning of the book?
I'm taking the overall meaning of the book as some sort of quest for redemption
I would agree that the overall theme of the book is a quest for redemption. Much of what Amir must redeem himself from involves his past, his father's past, and his country's past.
He must contend with his own cowardice and lack of loyalty to Hassan, which he finally does by fighting figuratively and literally for Hassan's son. He must also contend with the spectre of Assef, who, if you recall is the sociopath that Hassan protected Amir from, but whom Amir did not protect Hassan from. This also helps him to redeem himself.
Insofar as Baba is concerned, Amir redeems his relationship with him, albeit after Baba's death. This takes place on a few different levels. He finds he must contend with the "theft" which Baba perpetuated against Ali, by having sex with Ali's wife and hiding his transgression by passing Hassan off as Ali's son. Once Amir learns of this, it is up to him to make things right, which proves to be a most difficult task. Amir must also grapple with his feelings that he was not a worthy son, feelings that are exacerbated when he learns that Hassan was Baba's son, too.
Insofar as his country of birth is concerned, Amir must redeem himself by overcoming the prejudices with which he was raised, prejudices arising from the treatment of different ethnic groups in Afghanistan, and prejudices which figured in his own attitude toward Hassan and Ali. Even when Amir returns to Afghanistan, he finds there is still considerable prejudice against Hassan's son because of his ethnic group, and even worse, this prejudice is exhibited when Amir takes Sohrab home to America, where the Afghan community still turns its collective nose up at the Zahari.
Certainly, if Amir had had an easy relationship with his past, there would be no story at all. The entire novel is a quest for redemption, and also a quest for self-knowledge and self-actualization. The entire story is a good example of "the sins of the fathers" being visited upon the sons, and Amir's struggles involve not only his past, but also the past of his father and of his native land.