Chemical vapor deposition (CVD) is an essential method of depositing and fabricating large-area and high-quality graphene. In this work, molecular dynamics (MD) simulation technology is adopted to simulate the fabrication of graphene on the copper (111) crystal surface by chemical vapor deposition method. In order to eliminate the adverse effects of traditional MD method, an adapted potential system between carbon and copper atoms is introduced into the modeling of deposition and growth simulation of graphene. The results reveal the microscale growth mechanism of the graphene depositing on Cu(111) crystal surfaces, and the influence of temperature and carbon deposition rate (CDR) on the quality of graphene. The simulation results indicate that the deposition and growth of graphene consists of two stages. The first stage is to form binary carbons, trinary carbons and carbon chains. The second stage is to form carbon rings and the defects healing. The research results also reveal that high temperature can provide the carbon atoms with sufficient energy, which can help the carbon atoms to skip the energetic barrier between the two stages, and then achieve the deposition and growth of graphene. Moreover, the influence of temperature and carbon deposition rate are investigated in detail. The temperature mainly affects the defects and the flatness of graphene. The defects of graphene are the least and the surface can become the flattest at a deposition temperature of 1300 K. Higher temperature can cause the carbon atoms to irregularly move, and lower temperature can suppress the catalysis of the copper substrate. Both the higher and lower temperature can degrade the quality of the graphene surface. The CDR can influence the defects of graphene in growth. The lower value of CDR can lead to local growth on the graphene surface because of the lower nucleation density while the higher CDR is also able to cause the defects to form because of the uneven free energy distribution on the copper surface that has thermal fluctuation. It is shown that graphene can present the flattest surface when the value of CDR is set to be 5 ps^–1. According to the simulation process of deposition, it validates that the bi-layer and multi-layer graphene may grow based on the deposition of original single layer of graphene. As to the deposition and growth practice, it is suggested that the temperature 1300K should be suitable for the graphene CVD process of Cu(111) surface. The results in this work can provide a reference for understanding and implementing the fabrication of graphene on the Cu substrate by CVD methods.