Integrating knowledge graphs (KGs) to enhance the reasoning capabilities of large language models (LLMs) is an emerging research challenge in claim verification. While KGs provide structured, semantically rich representations well-suited for reasoning, most existing verification methods rely on unstructured text corpora, limiting their ability to effectively leverage KGs. Additionally, despite possessing strong reasoning abilities, modern LLMs struggle with multi-step modular pipelines and reasoning over KGs without adaptation. To address these challenges, we propose ClaimPKG, an end-to-end framework that seamlessly integrates LLM reasoning with structured knowledge from KGs. Specifically, the main idea of ClaimPKG is to employ a lightweight, specialized LLM to represent the input claim as pseudo-subgraphs, guiding a dedicated subgraph retrieval module to identify relevant KG subgraphs. These retrieved subgraphs are then processed by a general-purpose LLM to produce the final verdict and justification. Extensive experiments on the FactKG dataset demonstrate that ClaimPKG achieves state-of-the-art performance, outperforming strong baselines in this research field by 9%-12% accuracy points across multiple categories. Furthermore, ClaimPKG exhibits zero-shot generalizability to unstructured datasets such as HoVer and FEVEROUS, effectively combining structured knowledge from KGs with LLM reasoning across various LLM backbones.