Bloodshed is inherent in The Kite Runner, from the first time Amir watches a man die in the Buzkashi tournament to his later experience with battling kites. Amir is established to be a bookish, quiet boy, and the kite battle is his first sign of "masculine," violent interest. Battle kites have sharp strings and cut the hands of their operators:
I sliced a bright yellow kite with a coiled white tail. It cost me another gash on the index finger and blood trickled down into my palm. I had Hassan hold the string and sucked the blood dry, blotted my finger against my jeans.
Hassan runs to get the kites which fall -- he is the titular Kite Runner -- and he has shared in the metaphorical bloodshed of Amir's victory. However, when Hassan is attacked and sexually assaulted by bullies, Amir finds that he is unable to intervene:
I stopped watching, turned away from the alley. Something warm was running down my wrist. I blinked, saw I was still biting down on my fist, hard enough to draw blood from the knuckles. I realized something else. I was weeping.
I could step into that alley, stand up for Hassan--the way he’d stood up for me all those times in the past--and accept whatever would happen to me. Or I could run.
In the end, I ran.
But, always, my mind returned to the alley. To Hassan’s brown corduroy pants lying on the bricks. To the droplets of blood staining the snow dark red, almost black.
(Hosseini, The Kite Runner, Google Books)
Amir's blood, the blood he spilled to win the kite and that led Hassan to assault, is the same blood that he draws while trying to decide how to act. Amir cannot reconcile his earlier, jubilant bloodshed with this later, fearful bloodshed; throughout the novel, he feels shame for his cowardice and equates his own blood with Hassan's. Amir followed his own selfish motivations, and failed to help his friend -- later discovered to be his half-brother -- and now finds that he cannot take joy in his victory, or in any sort of bloodshed. Hassan's blood, then, is Amir's blood.