We present a lattice QCD determination of the $B_c \rightarrow J/\psi$ vector and axial-vector form factors over the full physical $q^2$ range with non-perturbatively renormalised lattice currents. Our calculation uses the Highly Improved Staggered Quark (HISQ) action on the second generation MILC gluon ensembles including light, strange and charm sea quarks. We use HISQ heavy quarks, with a range of masses going up to the $b$ on our finest lattices. Finally we use these form factors to compute the differential decay rates and construct the ratio of decay rates $R(J/\psi) = \Gamma(B_c \rightarrow J/\psi \tau^-\overline{\nu}_\tau)/\Gamma(B_c \rightarrow J/\psi \ell^-\overline{\nu}_\ell)$ where $\ell$ is either a muon or electron.