The pure and carbon (C)-doped $MgB_2$ thin films were fabricated on $Al_2O_3$ (0001) substrates at a temperature of $650^{circ}C$ by using hot-filament-assisted hybrid physical-chemical vapor deposition technique. The $T_c$ value for pure $MgB_2$ film is 38.5 K, while it is between 30 and 35 K for carbon-doped $MgB_2$ films. Expansion in c-axis lattice parameter was observed with increase in carbon doping concentration which is in contrast to carbon-doped $MgB_2$ single crystals. Significant enhancement in the critical current density was obtained for C-doped $MgB_2$ films as compared to the undoped $MgB_2$ film. This enhancement is most probably due to the incorporation of C into $MgB_2$ and the high density of grain boundaries, both help in the pinning of vortices and result in improved superconducting performance.