Chinese Named Entity Recognition (CNER) is a widely used technology in various applications. While recent studies have focused on utilizing additional information of the Chinese language and characters to enhance CNER performance, this paper focuses on a specific aspect of CNER known as fine-grained CNER (FG-CNER). FG-CNER involves the use of hierarchical, fine-grained categories (e.g. Person-MovieStar) to label named entities. To promote research in this area, we introduce the FiNE dataset, a dataset for FG-CNER consisting of 30,000 sentences from various domains and containing 67,651 entities in 54 fine-grained flattened hierarchical categories. Additionally, we propose SoftFiNE, a novel approach for FG-CNER that utilizes a custom-designed relevance scoring function based on label structures to learn the potential relevance between different flattened hierarchical labels. Our experimental results demonstrate that the proposed SoftFiNE method outperforms the state-of-the-art baselines on the FiNE dataset. Furthermore, we conduct extensive experiments on three other datasets, including OntoNotes 4.0, Weibo, and Resume, where SoftFiNE achieved state-of-the-art performance on all three datasets.