This paper studies a deep learning (DL) framework for the design of binary modulated visible light communication (VLC) transceiver with universal dimming support. The dimming control for the optical binary signal boils down to a combinatorial codebook design so that the average Hamming weight of binary codewords matches with arbitrary dimming target. An unsupervised DL technique is employed for obtaining a neural network to replace the encoder-decoder pair that recovers the message from the optically transmitted signal. In such a task, a novel stochastic binarization method is developed to generate the set of binary codewords from continuous-valued neural network outputs. For universal support of arbitrary dimming target, the DL-based VLC transceiver is trained with multiple dimming constraints, which turns out to be a constrained training optimization that is very challenging to handle with existing DL methods. We develop a new training algorithm that addresses the dimming constraints through a dual formulation of the optimization. Based on the developed algorithm, the resulting VLC transceiver can be optimized via the end-to-end training procedure. Numerical results verify that the proposed codebook outperforms theoretically best constant weight codebooks under various VLC setups.