A four-component formulation of relativistic density functional theory is presented together with the details of its implemention using a G-spinor basis set. The technical features of this approach are compared to those found in the nonrelativistic density functional theory of quantum chemistry which employ scalar basis sets of Gaussian-type functions. Numerical results of the G-spinor expansion method are presented for a sequence of closed-shell atoms, and for a selection of relativistic density functionals, and are compared with finite difference benchmarks.