In The Kite Runner, how does Amir's childhood relationship with his father impact his adult personality and relationships?

This is a great question. It is absolutely clear, that, as
for all of us, Amir's childhood relationship with his father greatly impacts him in his
adult years. One central aspect that you have to focus on is how Amir is presented as
being so incredibly different to Baba. Note that this is something that causes Baba
himself to doubt his own paternity of Amir:

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"If I hadn't seen the doctor pull him out of my
wife with my own eyes, I'd never believe he's my
son."

Throughout his life,
Baba is described as a strong character who is eager to stand up and fight against
injustice to protect the weak. And yet note how Amir reacts to a number of key
situations: he is scared of Assef, and crucially does nothing to help Hassan and prevent
his rape. Likewise, when Baba and Amir flee Afghanistan, Amir tries to dissuade his
father from challenging the Russian soldier who wants to rape one of the women
travelling with them. Amir is constantly aware of how he does not measure up when
compared with his father, and this is something that Rahim Khan uses when Amir is an
adult to convince him to go back to Kabul and rescue
Sohrab:

"You
know," Rahim Khan said, "one time, when you weren't around, your father and I were
talking. And you know how he always worried about you in those days. I remember he said
to me, 'Rahim, a boy who won't stand up for himself becomes a man who can't stand up to
anything.' I wonder, is that what you've
become?"

It is crucially this
haunting recognition of how Amir has not lived up to Baba's larger than life personality
that drives him to recognise and accept this part of his legacy and to face Assef and
his fears by going to Kabul to rescue Sohrab, and thus dispel the ghosts from his own
past.