Non-orthogonal multiple access (NOMA) utilizing the power domain and advanced receiver has been considered as one promising multiple access technology for further cellular enhancements toward the 5th generation (5G) mobile communications system. Most of the existing investigations into NOMA focus on the combination of NOMA with orthogonal frequency division multiple access (OFDMA) for either downlink or uplink. In this paper, we investigate NOMA for uplink with single carrier-frequency division multiple access (SC-FDMA) being used. Differently from OFDMA, SC-FDMA requires consecutive resource allocation to a user equipment (UE) in order to achieve low peak to average power ratio (PAPR) transmission by the UE. Therefore, sophisticated designs of scheduling algorithm for NOMA with SC-FDMA are needed. To this end, this paper investigates the key issues of uplink NOMA scheduling such as UE grouping method and resource widening strategy. Because the optimal schemes have high computational complexity, novel schemes with low computational complexity are proposed for practical usage for uplink resource allocation of NOMA with SC-FDMA. On the basis of the proposed scheduling schemes, the performance of NOMA is investigated by system-level simulations in order to provide insights into the suitability of using NOMA for uplink radio access. Key issues impacting NOMA performance are evaluated and analyzed, such as scheduling granularity, UE number and the combination with fractional frequency reuse (FFR). Simulation results verify the effectiveness of the proposed algorithms and show that NOMA is a promising radio access technology for 5G systems.