With the fast development of Internet of Things (IoT), the fifth generation (5G) wireless networks need to provide massive connectivity of IoT devices and meet the demand for low latency. To satisfy these requirements, nonorthogonal multiple access (NOMA) has been recognized as a promising solution for 5G networks to significantly improve the network capacity. In parallel with the development of NOMA techniques, mobile edge computing (MEC) is becoming one of the key emerging technologies to reduce the latency and improve the quality of service (QoS) for 5G networks. In order to capture the potential gains of NOMA in the context of MEC, this paper proposes an edge computing aware NOMA technique which can enjoy the benefits of uplink NOMA in reducing MEC users' uplink energy consumption. To this end, we formulate an NOMA-based optimization framework which minimizes the energy consumption of MEC users via optimizing the user clustering, computing and communication resource allocation, and transmit powers. In particular, similar to frequency resource blocks (RBs), we divide the computing capacity available at the cloudlet to computing RBs. Accordingly, we explore the joint allocation of the frequency and computing RBs to the users that are assigned to different order indices within the NOMA clusters. We also design an efficient heuristic algorithm for user clustering and RBs allocation, and formulate a convex optimization problem for the power control to be solved independently per NOMA cluster. The performance of the proposed NOMA scheme is evaluated via simulations.