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Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative temperatures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as Tc2 = 0.9565(7). Investigating the fixed-point tensor, we precisely locate the transition temperatures at Tc1 = 0.9029(1) and Tc2 = 0.9520(1), consistent well with the Monte Carlo and the density matrix renormalization group results. [ABSTRACT FROM AUTHOR]