This paper proposes a Cluster-aware Graph Contrastive Learning Framework (CueGCL) that simultaneously performs cluster information learning and node representation learning to address the performance degradation of existing graph contrastive learning (GCL) algorithms in structure-related unsupervised learning tasks such as graph clustering. Specifically, we design a personalized self-learning (PeST) strategy for unsupervised learning scenarios to capture accurate cluster-level personalized information and mitigate class conflicts and unfairness. Furthermore, we demonstrate that aligned graph clustering (AGC) yields consistent node embeddings and, theoretically demonstrating its effectiveness, generates an embedding space with clearly distinct cluster structures. Experimental results on five benchmark datasets demonstrate that CueGCL achieves state-of-the-art performance.