In this work, the DC Josephson effect is investigated for a superconductor-graphene-superconductor junction in both short- and long-junction regimes. The electric transport properties are calculated while taking into account the contribution of the discrete and continuous energy spectrum. In our approach, the phase dependence of the critical current is calculated at arbitrary temperature and doping level, which generalizes previous results. We show that critical current ${I}_{c}$ and skewness $S$ exhibit critical points as a function of graphene doping ${E}_{F}$, which can be explained by Klein resonances in graphene. We give a general characterization of $S$ vs ${I}_{c}$ curves while fixing temperature or doping level. When the temperature dependence of ${I}_{c}$ is analyzed, we find differences with respect to conventional Josephson junctions, given that there is a relevant doping effect. In the long-junction regime with ${E}_{F}$ far away from the Dirac point, the ${I}_{c}$ vs $T$ curve may exhibit an exponential decay law, which has been measured recently. We report the temperature dependence of $S$ in the whole range of temperature, and our approach allows us to account for skewness suppression in the vicinity of the Dirac point, which is in agreement with recent experiments. We mention some effects which can be attained in Josephson junctions with well-defined edges and for transparency values below unity of the graphene-superconductor interfaces.