Kolmogorov complexity applications in computational complexity
Lance Fortnow has written an article on this topic: Kolmogorov Complexity and Computational Complexity

You should also check out An Introduction to Kolmogorov Complexity and Its Applications by Li and Vitanyi, the definitive book on the subject.  In particular, chapter 6 "The Incompressibility Method" discusses a number of applications in complexity such as a Kolmogorov complexity proof of Hastad's switching lemma (from Circuit Lower Bounds à la Kolmogorov by Fortnow and Laplante).

And there are applications in communication complexity (e.g. Kolmogorov Complexity and Combinatorial Methods in Communication Complexity by Kaplan and Laplante).