  Inspired by the recent measurement of the ratio of $B_c$ branching fractions
to $J/\psi \pi^+$ and $J/\psi \mu^+\nu_{\mu}$ final states at the LHCb
detector, we study the semileptonic decays of $B_c$ meson to the S-wave ground
and radially excited 2S and 3S charmonium states with the perturbative QCD
approach. After evaluating the form factors for the transitions $B_c\rightarrow
P,V$, where $P$ and $V$ denote pseudoscalar and vector S-wave charmonia,
respectively, we calculate the branching ratios for all these semileptonic
decays. The theoretical uncertainty of hadronic input parameters are reduced by
utilizing the light-cone wave function for $B_c$ meson. It is found that the
predicted branching ratios range from $10^{-6}$ up to $10^{-2}$ and could be
measured by the future LHCb experiment. Our prediction for the ratio of
branching fractions $\frac{\mathcal {BR}(B_c^+\rightarrow J/\Psi
\pi^+)}{\mathcal {BR}(B_c^+\rightarrow J/\Psi \mu^+\nu_{\mu})}$ is in good
agreement with the data. For $B_c\rightarrow V l \nu_l$ decays, the relative
contributions of the longitudinal and transverse polarization are discussed in
different momentum transfer squared regions. These predictions will be tested
on the ongoing and forthcoming experiments.
