Which Afghan social classes do The Kite Runner's ...

This is a really good question and its answer in description, is quite long and historical.
In Afghanistan, socioeconomic status was highly correlated with ethnicity stratifying the greater Afghan society. Income inequality was vast as most of the upper class came from the royal tribal clan, while the lower class was comprised of the likes of Hassan's family of The Kite Runner.
Amir's father Baba (Homayoun Ershadi) is an aristocratic scholar and pundit, therefore, Amir represents the upper class. Hassans' is a Hazara. Hazara's were lower class and most, treated as servants and slaves. While slavery was outlawed from Afghanistan in the early 1900's, still until the early 1970s era when The Kite Runner's young Amir grows up in Kabul, the slave-like old practices of Hazaras were continued. While some had broken from this bondage, they were not treated much better earning low pay as servants ('nokahr' or 'muzdur') laboring as attendants, cooks, housekeepers, drivers, midwives, cloth washers, and yard workers in many middle-class to high-ranking households.
You can read deeper into the historical explanations of Afghanistan's social class here.