Using angle-resolved photoemission spectroscopy, we studied the electronic structure of NaFe$_{1-x}$Co$_x$As from an optimally doped superconducting compound ($x=0.028$) to a heavily overdoped non-superconducting one ($x=0.109$). Similar to the case of 122 type iron pnictides, our data suggest that Co dopant in NaFe$_{1-x}$Co$_x$As supplies extra charge carriers and shifts the Fermi level accordingly. In the $x=0.109$ compound, the hole-like bands around the zone center $Gamma$ move to deeper binding energies and an electron pocket appears instead. The overall band renormalization remains basically the same throughout the doping range we studied, suggesting that the local magnetic/electronic correlations are not affected by carrier doping. We speculate that a balance between itinerant properties of mobile carriers and local interactions may play an important role for the superconductivity.