Unsupervised graph representation learning aims to learn low-dimensional node embeddings without supervision while preserving graph topological structures and node attributive features. Previous graph neural networks (GNN) require a large number of labeled nodes, which may not be accessible in real-world graph data. In this paper, we present a novel cluster-aware graph neural network (CAGNN) model for unsupervised graph representation learning using self-supervised techniques. In CAGNN, we perform clustering on the node embeddings and update the model parameters by predicting the cluster assignments. Moreover, we observe that graphs often contain inter-class edges, which mislead the GNN model to aggregate noisy information from neighborhood nodes. We further refine the graph topology by strengthening intra-class edges and reducing node connections between different classes based on cluster labels, which better preserves cluster structures in the embedding space. We conduct comprehensive experiments on two benchmark tasks using real-world datasets. The results demonstrate the superior performance of the proposed model over existing baseline methods. Notably, our model gains over 7% improvements in terms of accuracy on node clustering over state-of-the-arts.