In the era of digital healthcare, biomedical data sharing is of paramount importance for the advancement of research and personalised healthcare. However, sharing such data while preserving user privacy and ensuring data security poses significant challenges. This paper introduces BioChainReward (BCR), a blockchain-based framework designed to address these concerns. BCR offers enhanced security, privacy, and incentivisation for data sharing in biomedical applications. Its architecture consists of four distinct layers: data, blockchain, smart contract, and application. The data layer handles the encryption and decryption of data, while the blockchain layer manages data hashing and retrieval. The smart contract layer includes an AI-enabled privacy-preservation sublayer that dynamically selects an appropriate privacy technique, tailored to the nature and purpose of each data request. This layer also features a feedback and incentive mechanism that incentivises patients to share their data by offering rewards. Lastly, the application layer serves as an interface for diverse applications, such as AI-enabled apps and data analysis tools, to access and utilise the shared data. Hence, BCR presents a robust, comprehensive approach to secure, privacy-aware, and incentivised data sharing in the biomedical domain.