Amir's birthday gifts are blood money because his lavish birthday party occurs shortly after his betrayal of Hassan. Hassan was his sacrifice for Baba's approval, and Amir cannot forget his selfishness and cowardice (In Chapter 7, he does nothing while Hassan defends his right to the kite and is subsequently raped by Assef). While Baba most likely would have thrown Amir some kind of party, Amir believes that the huge party--which includes all the guests bringing gifts--would not have occurred if he had not won the kite competition and brought home the last fallen kite to Baba (he only had the kite because of Hassan's loyalty to him). Thus, all of the gifts, Baba's approval, and even the party make Amir feel even guiltier for what he did.
The phrase "blood money" refers to something that one has gained by devious or harmful means. Amir gained Baba's approval from Hassan's literally spilling his own blood for him.
I a sense there are two meanings behind his reference to having the birthday gifts as blood money. In the book The Kite Runner, Amir wants his fathers love and support. He knows that he is a disappointment to Baba. He wries and reads his stories to his father, but nothing he does demonstrates to him that his father accepts him for who he is.
The day of the kite tournament is a turning point in the story. Amir and Hassan attend the kite tournament in Kabul. Amir for once demonstrates aggression during the tournament and wins by cutting the other kite's string. As a result he wins the tournament.
Baba finally sees some hope that his son can stand up and be aggressive. He showed he can be a winner. When he gives such magnificent presents to Amir, the boy knows that he got hem because he had won. He also recognizes that he is only appreciated by his father at this moment in time because he did win.
The second reason that it is blood money is because after the tournament Hassan had run for the kite. He ran into the alley and Assef and his friends accosted him. Hassan would not give Assef the kite which angered Assef. As a result Assef raped Hassan while Amir hid behind a wall evident of the events. He was unable to stand up for Hassan. As Hassan left he still had the kite and gave it to Amir. The blood from the rape that was shed by Hassan was the other reason those gifts felt like blood money.