In this talk I will present new techniques (inspired by work of Bendel-Nakano-Pillen) for computing cohomology for the finite Chevalley group G(F_q) directly in terms of cohomology for the ambient algebraic group G and its associated Frobenius kernels. These techniques will be used to compute the first and second cohomology group when in when M is a simple G(F_q)-module. A salient feature of our results is that we require no twisting of the coefficient module by the Frobenius morphism, which enables us to make calculations for relatively small values of p and q. Our calculations extends the seminal cohomological calculations of Cline-Parshall-Scott (1975), Jones (1975), Bell (1978), and Kleshchev (1994). This research was conducted by the University of Georgia VIGRE Algebra Group during the academic years 2009-10 and 2010-11.