We propose a deep learning-based channel estimation, quantization, feedback, and precoding method for downlink multiuser multiple-input and multiple-output systems. In the proposed system, channel estimation and quantization for limited feedback are handled by a receiver deep neural network (DNN). Precoder selection is handled by a transmitter DNN. To emulate the traditional channel quantization, a binarization layer is adopted at each receiver DNN, and the binarization layer is also used to enable end-to-end learning. However, this can lead to inaccurate gradients, which can trap the receiver DNNs at a poor local minimum during training. To address this, we consider knowledge distillation, in which the existing DNNs are jointly trained with an auxiliary transmitter DNN. The use of an auxiliary DNN as a teacher network allows the receiver DNNs to additionally exploit lossless gradients, which is useful in avoiding a poor local minimum. For the same number of feedback bits, our DNN-based precoding scheme can achieve a higher downlink rate compared to conventional linear precoding with codebook-based limited feedback.