Theorem 1.4.1 in [arxiv:0810.2076][1] answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We do not know about other results on the representation ring of $GL_n({\mathbb F}_q)$.


  [1]: http://arxiv.org/abs/0810.2076