  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 $B_c^-\rightarrow \eta_c +D^-(D^{*-}_0)$, which need about
$\sim10^7$ $B_c$ events in experiment. Decays to $ \eta_c +D^{*-}(D_s^-),
h_c+D^-(D^{*-})(D^{*-}_0), J/\Psi+D^{*-}$ are also hopeful channels.
