We present first-principles calculations of the nonlinear optical (NLO) response of a Ni/Cu(001) bilayer. The calculations are based on the full potential linearized augmented plane wave (FLAPW) method with the additional implementation of spin-orbit coupling (SOC). On the basis of this set of eigenstates the magneto-optical transition-matrix elements are evaluated. Using the surface-sheet model the optical reflection properties are determined for the cases of the magnetization vector perpendicular to the surface (polar magneto-optical configuration (MOC)) and for the in-plane magnetization (longitudinal MOC). The nonlinear optical susceptibility tensor elements <chi>⁽²⁾_ijk for different magnetization directions as well as the spectral dependence of <chi>⁽²⁾_ijk, the resulting intensities, and Kerr angles are presented for the Ni/Cu(001) bilayer. The results show that the magnetic tensor elements of the <chi>⁽²⁾_ijk tensor are smaller than the nonmagnetic ones by only one order of magnitude, confirming the important role of magnetic properties in the NLO response.