Summary: Rates and CP asymmetries of the non-leptonic two-body decay of \(B_{c}\) are calculated based on the low energy effective Hamiltonian. We concentrate on such \(b\) quark decays of the processes with \(0^{-}\) and \(1^{-}\) S-wave particles and/or \(0^{+}\) and \(1^{+}\) P-wave particles in the final states. The Salpeter method, which is the relativistic instantaneous approximation of the original Bethe-Salpeter equation, is used to derive hadron transition matrix elements. Based on the calculation, it is found that the best decay channels to observe CP violation are \(B_{c}^{-} \to \eta_{c} + D^{-}(D_{0}^{\ast -})\), which need about \(\sim 10^{7}B_{c}\) events in experiment. Decays to \(\eta_{c} + D^{\ast -}(D_{s}^{-})\), \(h_{c} + D^{-}(D^{\ast -})(D_{0}^{\ast -})\), \(J/{\Psi} + D^{\ast -}\) are also hopeful channels.