This area of the subject is somewhat frustrating, since there is a lot of literature but not many satisfactory results involving the entire cohomology ring. For what it's worth, I'll point you to a survey I wrote around 2005 (not up to date at present) here. See especially the short Chapter 14 and its many references. For groups of rank 1, see for example 14.7. [This survey was meant to be a supplement to Jantzen's large 1987 book Representations of Algebraic Groups, 2nd ed., AMS, 2003, but it got delayed many years.]
One person who can give a clearer view of the entire subject of cohomology for finite groups of Lie type is Dan Nakano at U. Georgia. But quite a few people have made contributions.