Molecular Dynamics simulation are employed to investigate the structures and mechanical behavior of both symmetric and asymmetric Σ5[0 0 1] tilt grain boundaries (GBs) of copper bicrystal under uniaxial tension and shear deformation. Simulation results indicate that the Σ5 asymmetric GBs can facet into their corresponding symmetric GB structures. The maximum tensile stress of symmetric GBs is higher than the asymmetric ones at both 10 K and 300 K, which suggests the symmetric GBs may have a more stable boundary structures. All the Σ5 GBs investigate in this study can migrate under the shear deformation with different velocity. The migration of Σ5 symmetric GBs is realized by uniform displacement of local atoms and rotation of the atomic group in E structural unit, while for the asymmetric GBs, the migration is identified to be a diffusion-related process result from local atoms shuffling.