The normal metal-quantum dots-superconductor hybrid system is a good platform for studying the mechanism of thermoelectric conversion. In terms of non-equilibrium Keldysh Green’s function formalism and linear response theory, the charge and spin thermoelectric transport characteristics of a normal-double quantum dot-superconductor hybrid system with spin-orbit coupling are studied in this work. We delve into the relationship between thermoelectric coefficients and the system parameters, and find both charge and spin thermoelectric coefficients exhibit distinct symmetry in the parameter space composed of temperature and energy. The increase in temperature leads to a decrease in conductance within the energy gap, which is attributed to the reduction in Andreev transport. However, outside the energy gap, the conductance gradually increases, and the thermal conductance is gradually enhanced. This is because more quasiparticles outside the energy gap participate in thermoelectric transport, and a large charge thermopower is generated in the region far from the energy gap. It is found that the thermoelectric figure of merit is greater than 1, indicating a strong violation of the Wiedemann-Franz law. With the increase of temperature, the large spin thermopower as well as spin thermoelectric figure of merit can be obtained outside the energy gap. The charge (spin) thermopower and the thermoelectric figure of merit show the rich evolutionary characteristics as functions of energy level and Zeeman energy. With the disappearance of the charge thermopower, the spin thermopower still has a finite value, which leads to the emergence of a pure spin Seebeck effect. This is helpful for designing a pure spin current thermoelectric generator. Due to a competitive mechanism between the spin-orbit coupling effect and the Zeeman field, thermoelectric coefficients decrease with the strength of spin-orbit interaction increasing, but one still can obtain the spin thermoelectric quantities which meet the practical needs by regulating the strength of spin-orbit coupling and the Zeeman energy. The evolution pattern of the thermoelectric coefficientss in the energy space indicates that the enhancement of thermoelectric conversion efficiency can be achieved by modulating the energy levels of double quantum dots. In addition, this hybrid system can act as a heat engine to achieve the conversion of heat into work. Although its power and efficiency do not evolve synchronously, thermodynamic performance that meets practical needs can still be obtained in certain parameter regions. The research results of this work hold theoretical and practical significance for understanding the thermoelectric transport and thermodynamic performance of hybrid thermoelectric systems.