The convergence of mobile edge computing (MEC) and blockchain is transforming the current computing services in mobile networks, by offering task offloading solutions with security enhancement empowered by blockchain mining. Nevertheless, these important enabling technologies have been studied separately in most existing works. This article proposes a novel cooperative task offloading and block mining (TOBM) scheme for a blockchain-based MEC system where each edge device not only handles data tasks but also deals with block mining for improving the system utility. To address the latency issues caused by the blockchain operation in MEC, we develop a new Proof-of-Reputation consensus mechanism based on a lightweight block verification strategy. A multi-objective function is then formulated to maximize the system utility of the blockchain-based MEC system, by jointly optimizing offloading decision, channel selection, transmit power allocation, and computational resource allocation. We propose a novel distributed deep reinforcement learning-based approach by using a multi-agent deep deterministic policy gradient algorithm. We then develop a game-theoretic solution to model the offloading and mining competition among edge devices as a potential game, and prove the existence of a pure Nash equilibrium. Simulation results demonstrate the significant system utility improvements of our proposed scheme over baseline approaches.