Considering the Debye-Hückel screening potential into Dirac equation, the plasma electron temperature and density effects on the energies of bound state 1s(2S1/2), 2s(2S1/2), 2p(2P1/2 and 2P3/2) for H-like ion C5+ are studied. A fitted formula, which estimates the bound state energies and ionization potential of H-like ion C5+ under different plasma conditions, is obtained. With this formula, the critical electron density of pressure ionization for bound levels is calculated, and agrees well with the data in previous literature. The results show that the energies of bound states increase with the increase of plasma electron densities or decrease of plasma electron temperatures. The relation between the logarithm of percentage shift of energy and the logarithm of plasma electron density or temperature is approximately linear. The results are useful in calculating the plasma ionization state distributions and plasma spectral simulations.