In this work, a method based on the traditional Gerchberg-Saxton (GS) algorithm and convolutional neural network (CNN) is proposed to generate vortex beams using a liquid crystal spatial light modulator (LC-SLM). By adopting this GS-CNN method, the Bessel beams with different topological charges are generated. On this basis, the root mean squared error (RMSE) and diffraction efficiency (DE) of the generated vortex beams are further analyzed and compared with the results obtained by the traditional GS algorithm. The results show that the GS-CNN method proposed in this paper can produce high-quality Bessel vortex beams. Compared with the results from the traditional GS algorithm, the intensity difference between the generated vortex beam and the target light is reduced and there are more input light field energies to be diffracted.