We have studied the ground state and thermodynamic properties of the anisotropic 3-state Potts model on square lattice by means of the tensor network-based numerical method. The phase diagrams of this model in the ground state and at finite temperature are identified. The singular behavior at the critical endpoint along the phase boundary is carefully investigated. It is discovered that the sharp peaks appear in the second-order derivative of the field as well as the first-order derivative of the magnetization with respect to temperature on the phase boundary. Our numerical results confirm the prediction of Fisher et al. about the critical endpoint.