Endowing chatbots with a consistent persona is particularly important to generate engaging dialogues. Existing models struggle to effectively perceive and comprehend persona, especially when confronted with persona tags that have unique and obscure meanings, presenting barriers to generating a consistent response. Besides, previous studies have not fully utilized the rich persona information implied in the dialogue history. In this paper, we propose a Persona-Aware chain-of-thought (CoT) learning framework for PErsonalized Response generation (PAPER), encompassing the persona understanding stage, the persona perception stage and the response generation stage. Specifically, we first leverage persona explanation data to train the model to interpret persona tags, equipping the model with ability to comprehend and interpret personas. Moreover, the model obtained during the persona understanding stage is also employed to extract persona information from the dialogue history and generate persona explanations to enhance persona traits. Subsequently, we utilize expanded persona information and dialogue history to generate consistent response. The generated persona explanations and the persona information extracted from the dialogue are regarded as intermediate persona CoT rationales. The response generation can leverage generated persona CoT rationales that are based on persona tags and dialogue history. Extensive experiments demonstrate that PAPER outperforms baselines and can effectively improve the quality and persona consistency of generated responses.