The nearest-neighbor resonating-valence bond (NNRVB) state is studied using classical anticommuting (Grassmann) variables. The classical partition function corresponding to the self-overlap of the NNRVB wavefunction is generated from a local (bond) Hamiltonian expressed in terms of four anticommuting variables. It is shown that the one-particle-per-site constraint introduces an interaction term which is a local product of all four variables. Two approaches are applied to study this Hamiltonian: (i) a self-consistent field decoupling scheme and (ii) a systematic perturbation expansion around the unconstrained soluble point. Bounds on the norm of the wavefunction are derived. Extensions to the presence of holes, long-range valence bonds, and the introduction of phase fluctuations [which violate the Marshall sign rule and yield a U(1) gauge theory] are discussed.