  We present a detailed analysis on the $B_c$ meson semi-leptonic decays, $B_c
\to \eta_c (J/\psi) \ell \nu$, up to next-to-leading order (NLO) QCD
correction. We adopt the principle of maximum conformality (PMC) to set the
renormalization scales for those decays. After applying the PMC scale setting,
we determine the optimal renormalization scale for the $B_c\to\eta_c(J/\psi)$
transition form factors (TFFs). Because of the same $\{\beta_0\}$-terms, the
optimal PMC scales at the NLO level are the same for all those TFFs, i.e.
$\mu_r^{\rm PMC} \approx 0.8{\rm GeV}$. We adopt a strong coupling model from
the massive perturbation theory (MPT) to achieve a reliable pQCD estimation in
this low energy region. Furthermore, we adopt a monopole form as an
extrapolation for the $B_c\to\eta_c(J/\psi)$ TFFs to all their allowable $q^2$
region. Then, we predict $\Gamma_{B_c \to \eta_c \ell \nu}(\ell=e,\mu)
=(71.53^{+11.27}_{-8.90})\times 10^{-15} {\rm GeV}$, $\Gamma_{B_c \to \eta_c
\tau \nu}=(27.14^{+5.93}_{-4.33})\times 10^{-15} {\rm GeV}$, $\Gamma_{B_c \to
J/\psi \ell \nu}(\ell=e,\mu) =(106.31^{+18.59}_{-14.01}) \times 10^{-15} {\rm
GeV}$, $\Gamma_{B_c \to J/\psi \tau \nu} =(28.25^{+6.02}_{-4.35})\times
10^{-15} {\rm GeV}$, where the uncertainties are squared averages of all the
mentioned error sources. We show that the present prediction of the production
cross section times branching ratio for $B^+_c\to J/\psi \ell^+ v$ relative to
that for $B^+ \to J/\psi K^+$, i.e. $\Re(J/\psi \ell^+ \nu)$, is in a better
agreement with CDF measurements than the previous predictions.
