Patent classification is a fundamental step in the patent examination process, directly impacting the efficiency and quality of substantive review. Existing methods mostly focus on general texts like titles and abstracts, thus ignoring the key technical content claims and the corresponding citation relationships. Meanwhile, these approaches treat labels as independent targets, failing to exploit the semantic and structural information within the label taxonomy. To address these problems, we propose a Claim Structure based Patent Classification model with Label Awareness (CSPC-LA). The method first utilizes the citation relationship of patent claim texts to construct the citation graph and the co-reference graph. Then structural graph learning is used on both graphs to mine the internal logic of patent claims. Finally, we optimize the tree hierarchy of IPC labels and employ tree propagation learning to enhance the patent representation. Extensive experiments on the latest patent classification dataset from USPTO demonstrate that the proposed method is more effective than the state-of-the-art baselines.