The Stanley chromatic symmetric polynomial X_G of a graph G is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the techniques of Khovanov homology to construct a homology of bigraded S_n-modules, whose bigraded Frobenius series reduces to the chromatic symmetric polynomial at q=t=1. We also obtain analogues of several familiar properties of the chromatic symmetric polynomial in terms of homology, including the decomposition formula for X_G discovered recently by Orellana and Scott, and Guay-Paquet.