  We make a systematic investigation on the two-body nonleptonic decays
$B_c\rightarrow J/\Psi(\eta_c),M$ by employing the perturbative QCD approach
based on $k_T$ factorization, where $M$ is a light pseudoscalar or vector or a
heavy charmed meson. We predict the branching ratios and direct CP asymmetries
of these $B_c$ decays and also the transverse polarization fractions of
$B_c\rightarrow J/\Psi V,J/\psi D^{*}_{(s)}$ decays. It is found that these
decays have a large branching ratios of the order of $10^{-4} - 10^{-2}$ and
could be measured by the future LHC-b experiment. Our predictions for the
ratios of branching fractions $\frac{\mathcal {BR}(B_c^+\rightarrow J/\Psi
D_s^+)} {\mathcal {BR}(B_c^+\rightarrow J/\Psi \pi^+)}$ ,$\frac{\mathcal
{BR}(B_c^+\rightarrow J/\Psi D_s^{*+})} {\mathcal {BR}(B_c^+\rightarrow J/\Psi
D_s^+)}$ and $\frac{\mathcal {BR}(B_c^+\rightarrow J/\Psi K^+)} {\mathcal
{BR}(B_c^+\rightarrow J/\Psi \pi^+)}$ are in good agreement with the data. A
large transverse polarization fraction which can reach $48\%$ is predicted in
$B_c^+\rightarrow J/\Psi D_s^{*+}$ decay, which is consistent with the data.
  We find a possible direct CP violation in $B_c\rightarrow J/\psi D^{*}$
decays, which are helpful to test the CP violating effects in $B_c$ decays.
