Cloud-radio access networks (C-RAN) help overcoming the scarcity of radio resources by enabling dense deployment of base-stations (BSs), and connecting them to a central-processor (CP). This paper considers the downlink of a C-RAN, where the cloud is connected to the BSs via limited-capacity backhaul links. We propose and optimize a C-RAN transmission scheme that combines rate splitting, common message decoding, beamforming vectors design and clustering. To this end, the paper optimizes a transmission scheme that combines rate splitting (RS), common message decoding (CMD), clustering and coordinated beamforming. In this work we focus on maximizing the weighted sum-rate subject to per-BS backhaul capacity and transmit power constraints, so as to jointly determine the RS-CMD mode of transmission, the cluster of BSs serving private and common messages of each user, and the associated beamforming vectors of each user private and common messages. The paper proposes solving such a complicated non-convex optimization problem using l0-norm relaxation techniques, followed by inner-convex approximations (ICA), so as to achieve stationary solutions to the relaxed non-convex problem. Numerical results show that the proposed method provides significant performance gain as compared to conventional interference mitigation techniques in C-RAN which simply treat interference as noise (TIN).