The Circular Electron-Positron Collider (CEPC) has high requirements for bunch charge, transverse emittance, and longitudinal length at the injector exit. Consequently, designing a high-performance electron gun and injector has become a challenge. To design an injector that meets the targets, many nonlinear and mutually coupled parameters need to be considered simultaneously. Therefore, we propose a method of searching in a high-dimensional parameter space using a multi-objective genetic algorithm to optimize the normalized transverse emittance and longitudinal bunch length, thus to maximize the performance of the electron gun. Since the full simulation of bunch transportaion with spacecharge effect is extremly time consuming, we adopted the deep Gaussian process as an surrogate model to solve high-dimensional parameter optimization problem. Through the analysis of key factors affecting the evolution of beam transverse and longitudinal phase space, a total of 16 geometric parameters and 10 beam element parameters have been determined in this paper. we present a design optimization for an injector consisting of an L-band radio frequency electron gun, a pair of solenoids, and a traveling wave tube, with an initial charge of 10 nC. After calculating 8000 effective solutions, we acquired a good approximation to the Pareto front between two objectives. The corresponding transverse normalized emittance is 19.8 π·mm·mrad, and the RMS beam length is 1.0 mm. Compared with the design requirement, the transverse normalized emittance is reduced by about 70%.