We use the framework of a relativistic constituent quark model to study the semileptonic transitions of the B-c meson into ((c) over barc) charmonium states where ((c) over barc)=eta(c)(S-1(0)), J/psi(S-3(1)), chi(c0)(P-3(0)), chi(c1)(P-3(1)), h(c)(P-1(1)), chi(c2)(P-3(2)), psi(D-3(2)). We compute the q(2) dependence of all relevant form factors and give predictions for their semileptonic B-c decay modes including also their tau modes. We derive a formula for the polar angle distribution of the charged lepton in the (l nu(l))-c.m. frame and compute the partial helicity rates that multiply the angular factors in the decay distribution. For the discovery channel B-c -> J/psi(->mu(+)mu(-))l nu we compute the transverse/longitudinal composition of the J/psi which can be determined by an angular analysis of the decay J/psi ->mu(+)mu(-). We compare our results with the results of other calculations.
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