Assef is one of the truly perverted and villainous characters in The Kite Runner, by Khaled Hosseini. He is a bully and a bigot, prejudiced against the Hazara, like Hassan, which he scorns as "flat-noses." Assef is a proponent of Hitler and wants an ethnic cleansing in Afghanistan much like Hitler wanted in Germany. (Ironically, Assef's mother is German and she is appalled at her son's idea; Hassan has blond hair and blue eyes, unlike most Pashtuns.)
You ask whether Assef is Pashtun or Taliban, and the answer is both. His ethnicity is Pashtun, which he believes to be the superior ethnicity, and he also becomes a member of the Taliban--which is made up primarily of Pashtuns.
In chapter five of the novel, Assef and a few of his buddies stop and harass Amir and Hassan. This is when he shares his thoughts that the new leader of Afghanistan, who is apparently a close friend of Assef's family, should eradicate those whom Assef considers to be less desirable. In addition to insulting them, this is the "vision" Assef shares with Amir and Hassan:
“Afghanistan is the land of Pashtuns. It always has been, always will be. We are the true Afghans, the pure Afghans, not this Flat-Nose here [Hassan]. His people pollute our homeland, our watan. They dirty our blood.” He made a sweeping, grandiose gesture with his hands. “Afghanistan for Pashtuns, I say. That’s my vision.”
Assef shifted his gaze to me again. He looked like someone coming out of a good dream. “Too late for Hitler,” he said. “But not for us.”
This quote makes it clear that Assef is a Pashtun, and later we know that he is part of the Taliban.