In this paper, we develop a new framework called blockchain-based Radio Frequency (RF)-powered backscatter cognitive radio network. In the framework, IoT devices as secondary transmitters transmit their sensing data to a secondary gateway by using the RF-powered backscatter cognitive radio technology. The data collected at the gateway is then sent to a blockchain network for further verification, storage and processing. As such, the framework enables the IoT system to simultaneously optimize the spectrum usage and maximize the energy efficiency. Moreover, the framework ensures that the data collected from the IoT devices is verified, stored and processed in a decentralized but in a trusted manner. To achieve the goal, we formulate a stochastic optimization problem for the gateway under the dynamics of the primary channel, the uncertainty of the IoT devices, and the unpredictability of the blockchain environment. In the problem, the gateway jointly decides (i) the time scheduling, i.e., the energy harvesting time, backscatter time, and transmission time, among the IoT devices, (ii) the blockchain network, and (iii) the transaction fee rate to maximize the network throughput while minimizing the cost. To solve the stochastic optimization problem, we then propose to employ, evaluate, and assess the Deep Reinforcement Learning (DRL) with Dueling Double Deep Q-Networks (D3QN) to derive the optimal policy for the gateway. The simulation results clearly show that the proposed solution outperforms the conventional baseline approaches such as the conventional Q-Learning algorithm and non-learning algorithms in terms of network throughput and convergence speed. Furthermore, the proposed solution guarantees that the data is stored in the blockchain network at a reasonable cost.