The study of the statistical properties of collisionless systems formed by neutral particles subject to gravitational field represents an intriguing theoretical issue. In astrophysics, it directly relates to the description of collisionless gravitating dark matter (DM) halos. Structures of this type are expected to be characterized by intrinsically non-Maxwellian kinetic distribution functions (KDFs) and to exhibit temperature anisotropy, i.e. an anisotropy in the directional particle velocity dispersions. In this paper, a theoretical analysis of the issue is proposed, based on the kinetic theory developed in the framework of the Vlasov-Poisson description for nonrelativistic DM systems at equilibrium. By implementing the method of invariants, explicitsolutions for the equilibrium KDFs are constructed and expressed through generalized Gaussian distributions. A perturbative theory is developed which allows them to be cast in terms of Chapman-Enskog series representations and to evaluat