The unsupervised representation learning of the multiplex network has attracted significant attention due to its powerful ability to model multiple relation types between nodes. Existing methods generally learn representations with the original graph structure but disregard the fact that the prior graph structure is inevitably incomplete, which may mislead the feature aggregation process. Therefore, improving the graph structure is of importance for multiplex network unsupervised representation. Unfortunately, heterogeneous graph structure optimization in multiplex networks is still less explored and encounters the challenges of how to jointly consider multiple relation types to supplement the missing edges in each relation. To address this issues, we formulate each relation type as one view and propose the Multiplex network Unsupervised Representation model with Adaptive multi-view graph structure learning (MURA). To improve the graph structure, we develop a multi-view low-rank self-representation tensor learning module to optimize the heterogeneous graph structure which well explores the global community property. Besides, a local neighborhood manifold structure learning module is employed to capture the essential local connection. With the refined multiplex graph, we further propose a similarity regularization term to enhance the multiplex node representations. Comprehensive experiments on three downstream tasks confirm the superiority of our MURA among five datasets when compared to the state-of-the-art rivals.