This work addresses the joint design of fronthaul and edge links for a cache-aided cloud radio access network (C-RAN) system with a wireless fronthaul link. Motivated by the fact that existing techniques, such as C-RAN and edge caching, come at the cost of increased energy consumption, an energy efficiency (EE) metric is defined and adopted as the performance metric for optimization. As the fronthaul links can be used to transfer quantized and precoded baseband signals or hard information of uncached files, both soft- and hard-transfer fronthauling strategies are considered. Extensive numerical results validate the impact of edge caching, as well as the advantages of the energy-efficient design over the spectrally-efficient scheme. Additionally, the two fronthauling strategies—the soft- and hard-transfer schemes—are compared in terms of EE.