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Envisioned as a promising component of the future wireless Internet-of-Things (IoT) networks, the non-orthogonal multiple access (NOMA) technique can support massive connectivity with a significantly increased spectral efficiency. Cooperative NOMA is able to further improve the communication reliability of users under poor channel conditions. However, the conventional system design suffers from several inherent limitations and is not optimized from the bit error rate (BER) perspective. In this paper, we develop a novel deep cooperative NOMA scheme, drawing upon the recent advances in deep learning (DL). We develop a novel hybrid-cascaded deep neural network (DNN) architecture such that the entire system can be optimized in a holistic manner. On this basis, we construct multiple loss functions to quantify the BER performance and propose a novel multi-task oriented two-stage training method to solve the end-to-end training problem in a self-supervised manner. The learning mechanism of each DNN module is then analyzed based on information theory, offering insights into the proposed DNN architecture and its corresponding training method. We also adapt the proposed scheme to handle the power allocation (PA) mismatch between training and inference and incorporate it with channel coding to combat signal deterioration. Simulation results verify its advantages over orthogonal multiple access (OMA) and the conventional cooperative NOMA scheme in various scenarios.