In this paper we study hypersurfaces with the mean curvature functionH satisfying $\langle \nabla H, \nabla H\rangle $ in a Minkowski space of arbitrary dimen-sion. First, we obtain some conditions satised by connection forms ofbiconservative hypersurfaces with the mean curvature function whosegradient is light-like. Then, we use these results to get a classication ofbiharmonic hypersurfaces. In particular, we prove that if a hypersurfaceis biharmonic, then it must have at least 6 distinct principal curvaturesunder the hypothesis of having mean curvature function satisfying thecondition above.