The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials, including a tensor interaction under the spin and pseudospin symmetric limits. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number kappa. Some numerical results are also given, and the effect of tensor interaction on the bound states is presented. It is shown that tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under spin symmetric limit conditions in the absence of tensor interaction