We report Cu NMR and NQR studies of the cupric sulfide CuS from 4.2 K to room temperature. We have assigned the site for two distinct Cu-63 NMR lines based on an abrupt change of the Cu NMR spectrum at 55 K due to a structural phase transition for both the powder and the oriented CuS. We found that the Cu nuclear spin-lattice relaxation rate 1/T-1 is magnetic in origin and highly anisotropic, and that the temperature dependence of 1/T1T is similar to that of the uniform magnetic susceptibility chi(0)(T) The orbital contribution to the relaxation rate and the nature of the electron spin correlation at the Cu sites are discussed.