Cloud Radio Access Network (C-RAN) is becoming more than ever timely in 5G for dense network deployments, but its design has to be adapted to 5G requirements. To ensure high uplink throughput in cell-edge regions affected by inter-cell interference, C-RAN enables multiuser reception techniques among cells. In this paper, we propose an efficient end-to-end uplink transmission scheme dealing with implementation and deployment constraints on both communication interfaces in C-RAN, i.e., the wireless one between the users and the Remote Radio Heads (RRHs), and the fronthaul links between the RRHs and the central processing unit. Multi-cell non-orthogonal multiple access (NOMA) can improve spectral efficiency and cell-edge throughput, but its design needs to fit requirements regarding receiver complexity and delay. Optimizing the fronthaul rate allocation to maximize the benefit of the transmissions allows to exploit the fronthaul links effectively. Our model considers the throughput of NOMA transmissions in C-RAN and the expense related to fronthaul usage in various deployment scenarios. When the available fronthaul rate allows accurate transmission, optimizing fronthaul rate allocation results in 10% higher transmission benefit than uniform allocation. It also makes possible to involve less users in NOMA while keeping the benefit. The proposed strategy enables uplink multi-cell multiuser processing in a cost-effective manner in 5G C-RAN deployments.