A study of a model rod-like polyelectrolyte molecule immersed into a monovalent or divalent electrolyte is presented. Results from the hypernetted-chain/mean spherical approximation (HNC/MSA) theory, for inhomogeneous charged fluids, {ch are} compared with molecular dynamics (MD) simulations. As a particular case, the parameters of the polyelectrolyte molecule are mapped to those of a DNA molecule. An excellent qualitative, and in some cases quantitative, agreement between HNC/MSA and MD is found. Both, HNC/MSA and MD, predict the occurrence of overcharging, which is not present in the Poisson-Boltzmann theory. Mean electrostatic potential and local concentration profiles, $zeta$-potential and charge distribution functions are obtained and discussed in terms of the observed overcharging effect. Particularly interesting results are a very non-monotonic behavior of the $zeta$-potential, as a function of the rod charge density, and the overcharging by {em monovalent} counterions.
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