We measured the resistivity and magnetic susceptibility to map out the phase diagram of single crystalline NaFe$_{1-x}$Co$_x$As. Replacement of Fe by Co suppresses both the structural and magnetic transition, while enhances the superconducting transition temperature ($T_{rm c}$) and superconducting component fraction. Magnetic susceptibility exhibits temperature-linear dependence in the high temperatures up to 500 K for all the superconducting samples, but such behavior suddenly breaks down for the non-superconducting overdoped crystal, suggesting that the superconductivity is closely related to the T-linear dependence of susceptibility. Analysis on the superconducting-state specific heat for the optimally doped crystal provides strong evidence for a two-band s-wave order parameter with gap amplitudes of $Delta_1(0)/k_{rm B}T_{rm c}$= 1.78 and $Delta_2(0)/k_{rm B}T_{rm c}$=3.11, being consistent with the nodeless gap symmetry revealed by angle-resolved photoemission spectroscopy experiment.
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