For next generation mobile communications, Cloud-RAN (C-RAN) is an emerging network architecture to provide broadband services. C-RAN separates computation entities, i.e., Baseband Units (BBUs), from base stations (BSs) and puts BBUs in a cloud located in a centralized network. With C-RAN, UEs can receive data from multiple collaborative cells and thus can leverage the dynamic point selection (DPS) technology to improve network efficiency. When user equipments (UEs) enter a hotspot and can not be served due to congestion, data offloading from the hotspot to its neighboring cells may take place to balance heterogeneous cells' loads. This work shows how to integrate such DPS offloading with the Discontinuous Reception (DRX) mechanism, which allows UEs to turn off their radio interfaces in a periodical manner. We address the resource allocation problem in heterogeneously-loaded C-RAN by optimizing UEs' energy consumption based on DRX while reserving sufficient bandwidths for UEs considering their quality-of-service (QoS) through DPS. We propose an offloading-based DPS scheduling scheme by exploiting not only maximal instantaneous throughputs but also minimal energy cost. Simulation results show that our scheme can improve throughput, resource utilization, and energy consumption as compared to existing schemes.