Recently, graph contrastive learning (GCL) has emerged as one of the optimal solutions for node-level and supervised tasks. However, for structure-related and unsupervised tasks such as graph clustering, current GCL algorithms face difficulties acquiring the necessary cluster-level information, resulting in poor performance. In addition, general unsupervised GCL improves the performance of downstream tasks by increasing the number of negative samples, which leads to severe class collision and unfairness of graph clustering. To address the above issues, we propose a Cluster-aware Graph Contrastive Learning Framework (CueGCL) to jointly learn clustering results and node representations. Specifically, we design a personalized self-training (PeST) strategy for unsupervised scenarios, which enables our model to capture precise cluster-level personalized information. With the benefit of the PeST, we alleviate class collision and unfairness without sacrificing the overall model performance. Furthermore, aligned graph clustering (AGC) is employed to obtain the cluster partition, where we align the clustering space of our downstream task with that in PeST to achieve more consistent node embeddings. Finally, we theoretically demonstrate the effectiveness of our model, showing it yields an embedding space with a significantly discernible cluster structure. Extensive experimental results also show our CueGCL exhibits state-of-the-art performance on five benchmark datasets with different scales.