This is an outline of Quillen’s proof for the calculation of K-theory of finite fields, originally done by Quillen in , see also  for a slightly different presentations with more background materials included.
Here are the full post:
 Quillen, Daniel. “On the cohomology and K-theory of the general linear groups over a finite field.” Annals of Mathematics 96.3 (1972): 552-586.
 Mitchell, S. Notes on K theory of nite elds. Available online: