We propose in this paper a carrier-less amplitude and phase visible light communications (VLC) system with deep learning (DL)-based post-equalizer (EQ) to significantly increase the transmission data rate. The proposed system is analyzed for various conditions including modulation order, transmitted signal bandwidth, and non-line of sight VLC channel. Results show that the highest data rate and spectral efficiency of 100 Mb/s and 4.67 b/s/Hz are achieved for the modulation order and signal bandwidth of 64 and 25 MHz, respectively. In addition, we compare the performance and complexity of the proposed system with different types of EQs including least mean square and Volterra series. The study shows the DL-based EQ is qualified to mitigate mixed linear and nonlinear impairments by providing improved bit error rate performance compared to the other EQs for all modulation orders and the transmitted signal bandwidth.