We demonstrate a vapor cell atomic clock prototype based on continuous-wave (CW) interrogation and double-modulation coherent population trapping (DM-CPT) technique. The DM-CPT technique uses a synchronous modulation of polarization and relative phase of a bi-chromatic laser beam in order to increase the number of atoms trapped in a dark state, i.e. a non-absorbing state. The narrow resonance, observed in transmission of a Cs vapor cell, is used as a narrow frequency discriminator in an atomic clock. A detailed characterization of the CPT resonance versus numerous parameters is reported. A short-term frequency stability of $3.2 times 10^{-13} tau^{-1/2}$ up to 100 s averaging time is measured. These performances are more than one order of magnitude better than industrial Rb clocks and comparable to those of best laboratory-prototype vapor cell clocks. The noise budget analysis shows that the short and mid-term frequency stability is mainly limited by the power fluctuations of the microwave used to generate the bi-chromatic laser. These preliminary results demonstrate that the DM-CPT technique is well-suited for the development of a high-performance atomic clock, with potential compact and robust setup due to its linear architecture. This clock could find future applications in industry, telecommunications, instrumentation or global navigation satellite systems.