For current and future Internet of Things (IoT) networks, mobile edge-cloud computation offloading (MECCO) has been regarded as a promising means to support delay-sensitive IoT applications. However, offloading mobile tasks to the cloud is vulnerable to security issues due to malicious mobile devices (MDs). How to implement offloading to alleviate computation burdens at MDs while guaranteeing high security in mobile edge cloud is a challenging problem. In this paper, we investigate simultaneously the security and computation offloading problems in a multi-user MECCO system with blockchain. First, to improve the offloading security, we propose a trustworthy access control using blockchain, which can protect cloud resources against illegal offloading behaviours. Then, to tackle the computation management of authorized MDs, we formulate a computation offloading problem by jointly optimizing the offloading decisions, the allocation of computing resource and radio bandwidth, and smart contract usage. This optimization problem aims to minimize the long-term system costs of latency, energy consumption and smart contract fee among all MDs. To solve the proposed offloading problem, we develop an advanced deep reinforcement learning algorithm using a double-dueling Q-network. Evaluation results from real experiments and numerical simulations demonstrate the significant advantages of our scheme over existing approaches.