Contact tracing has been proven an effective approach to control the virus spread in pandemics like COVID-19 pandemic. As an emerging powerful decentralized technique, blockchain has been explored to ensure data privacy and security in contact tracing processes. However, existing works are mostly high-level designs with no sufficient demonstration and treat blockchain as separate storage system assisting third-party central servers, ignoring the importance and capability of consensus mechanism and incentive mechanism. In this paper, we propose a light-weight and fully third-party free Blockchain-Driven Contact Tracing framework (BDCT) to bridge the gap. In the BDCT framework, RSA encryption based transaction verification method (RSA-TVM) is proposed to ensure contact tracing correctness, which can achieve more than 96\% contact cases recording accuracy even each person has 60\% probability of failing to verify the contact information. Reputation Corrected Delegated Proof of Stake (RC-DPoS) consensus mechanism is proposed together with the incentive mechanism, which can ensure timeliness of reporting contact cases and keep blockchain decentralized. A novel contact tracing simulation environment is created, which considers three different contact scenarios based on population density. The simulation results demonstrate the effectiveness, robustness and attack resistance of RSA-TVM and RC-DPoS in the proposed BDCT.