Multiplex graph representation learning has attracted considerable attention due to its powerful capacity to depict multiple relation types between nodes. Previous methods generally learn representations of each relation-based subgraph and then aggregate them into final representations. Despite the enormous success, they commonly encounter two challenges: 1) the latent community structure is overlooked and 2) consistent and complementary information across relation types remains largely unexplored. To address these issues, we propose a clustering-enhanced multiplex graph contrastive representation learning model (CEMR). In CEMR, by formulating each relation type as a view, we propose a multiview graph clustering framework to discover the potential community structure, which promotes representations to incorporate global semantic correlations. Moreover, under the proposed multiview clustering framework, we develop cross-view contrastive learning and cross-view cosupervision modules to explore consistent and complementary information in different views, respectively. Specifically, the cross-view contrastive learning module equipped with a novel negative pairs selecting mechanism enables the view-specific representations to extract common knowledge across views. The cross-view cosupervision module exploits the high-confidence complementary information in one view to guide low-confidence clustering in other views by contrastive learning. Comprehensive experiments on four datasets confirm the superiority of our CEMR when compared to the state-of-the-art rivals.