Quillen's first construction of higher algebraic $K$-theory (the plus construction) was purely topological. It is related to group cohomology as mentioned in the question. The later constructions (Quillen's $Q$-construction, Waldhausen $K$-theory etc.) involve manipulations with categories, but in the end one always takes the homotopy groups of the classifying space. I am not sure if there is a construction now that throws out all topology. See Weibel's book for more details.