Autonomous electric vehicles (AEVs) are increasingly recognized as a crucial component in the modern transportation networks. This creates a pressing need to develop efficient and economic management methods for AEV operators. In this paper, we propose a coordinated order scheduling and charging framework for AEVs that provide ride-hailing services based on deep reinforcement learning (DRL) method that considers AEVs' spatial-temporal mobility and charging behaviors. In this framework, we first formulate an aggregated model for charging stations (CSs) that extracts features of each AEV by tracking the dynamic correspondence between AEVs and CSs. Then, we propose a novel DRL-based algorithm for the aggregated model that schedules the lower thresholds of charging requirements so that AEVs can get charged at the lowest charging prices and shortest trip distances, in the meantime satisfying ride-hailing order demand. Numerical experiments are conducted to verify the superiority of the proposed method through a New York City zone-based power-transportation coupled network. The results validate the effectiveness of the proposed framework in saving operation cost with high computational efficiency.