I believe an alternative condition for a kahler structure is having the covariant derivative of J vanish (with respect to the levi civita condition of your herimitian metric). Dunno how helpful that is. I'm not sure of any topological characterization myself but I do know that a *necessary* condition is that H^p,q (X)=H^q,p (X) (this follows from hodge theory) and in part H^i (X) is a vector space of even dimension for i odd. This allows you to construct counterexamples of cx manifolds which arent kahler
