I believe that one reason author Khaled Hosseini decides to make use of personal letters in The Kite Runner is because Hassan could not read or write prior to Amir leaving for California. Due to his undying admiration for Amir and the love he has for his old friend's writing and story-telling talents, Hassan decides to learn to read and write as well.
... he had befriended an old Farsi teacher in Bamiyan who had taught him to read and write. (Chapter 16)
Hassan probably chooses to become literate so he can write Amir letters in the hope that he will receive letters in return. Although Hassan dies before Rahim Khan can deliver the letters, they eventually fall into Amir's hands--just as Hassan would have wanted. And Amir is proud of his old friend's perseverance:
It was written in Farsi. No dots were omitted, no crosses forgotten, no words blurred together--the handwriting was almost childlike in its neatness. (Chapter 17)
The use of letters also provides Hosseini, through Hassan, with a way to show the goodness of Rahim Khan as well as the severity of his illness.