Faced with the massive connection, sporadic transmission, and small-sized data packets in future cellular communication, a grant-free non-orthogonal random access (NORA) system is considered in this paper, which could reduce the access delay and support more devices. In order to address the joint user activity detection (UAD) and channel estimation (CE) problem in the grant-free NORA system, we propose a deep neural network-aided message passing-based block sparse Bayesian learning (DNN-MP-BSBL) algorithm. In this algorithm, the message passing process is transferred from a factor graph to a deep neural network (DNN). Weights are imposed on the messages in the DNN and trained to minimize the estimation error. It is shown that the weights could alleviate the convergence problem of the MP-BSBL algorithm. Simulation results show that the proposed DNN-MP-BSBL algorithm could improve the UAD and CE accuracy with a smaller number of iterations.