By work of Waldhausen, Quillen and others, K-theory can be defined in quite abstract settings. It would be nice to understand this.
arXiv: Experimental full text search
KT (K-theory), CT (Category theory), AG (Algebraic Geometry)
Neeman: http://www.math.uiuc.edu/K-theory/0029, http://www.math.uiuc.edu/K-theory/0030
Less interesting: Yao
For K-theory of DG-categories, see Toen: Lecture on DG-categories. File Toen web unpubl swisk.pdf. Treats basic theory, localization, relation to model cats, functorial cones, K-theory and Hochschild cohomology, and descent problems.
nLab page on K-theory of categories