A 2NN MEAM (second nearest-neighbor modified embedded atomic method) potential for the copper and carbon atom interaction is developed. Since crystal structures like B1 or B2 are not experimentally available for the Cu-C system, first-principle calculations are used to determine the reference structure and its elastic constants in this work. The B1 and B2 structure of Cu-C has positive formation energy, but B1 is dynamically stable. Accordingly, the B1 crystal structure is used as the reference structure for the Cu-C system to develop the interatomic potential. The current potential agrees reasonably well for phonon dispersion frequencies, stacking fault energies, and the atomic forces with the available experimental data and first-principle calculations. This potential is utilized to study the mechanical properties of copper-graphene composites subjected to uniaxial loading. Our results show that adding graphene to a defect-free Cu crystal weakens the metallic matrix's mechanical properties. However, when the graphene is embedded into a Cu matrix with some defects, it can significantly improve the polycrystalline Cu's mechanical properties.