The unmanned aerial vehicle (UAV) plays an vital role in various applications such as delivery, military mission, disaster rescue, communication, etc., due to its flexibility and versatility. This paper proposes a deep reinforcement learning method to solve the UAV online routing problem with wireless power transfer, which can charge the UAV remotely without wires, thus extending the capability of the battery-limited UAV. Our study considers the power consumption of the UAV and the wireless charging process. Unlike the previous works, we solve the problem by a designed deep neural network. The model is trained using a deep reinforcement learning method offline, and is used to optimize the UAV routing problem online. On small and large scale instances, the proposed model runs from four times to 500 times faster than Google OR-tools, the state-of-the-art combinatorial optimization solver, with identical solution quality. It also outperforms different types of heuristic and local search methods in terms of both run-time and optimality. In addition, once the model is trained, it can scale to new generated problem instances with arbitrary topology that are not seen during training. The proposed method is practically applicable when the problem scale is large and the response time is crucial.