A novel dynamic radio-cooperation strategy is proposed for a Cloud Radio Access Network (Cloud-RAN) consisting of multiple Remote Radio Heads connected to a central Virtual Base Station (VBS) pool. In particular, the key capabilities of Cloud-RAN in computing-resource sharing and real-time communication among the VBSs are leveraged to design a joint dynamic radio clustering and cooperative beamforming scheme that maximizes the downlink Weighted Sum-Rate System Utility (WSRSU). Due to the combinatorial nature of the radio clustering process and to the non-convexity of the cooperative beamforming design, the underlying optimization problem is NP-hard, and is extremely difficult to solve for a large network. The proposed approach aims for a suboptimal solution by transforming the original problem into a Mixed-Integer Second-Order Cone Program (MI-SOCP) and applying Sequential Convex Approximation (SCA) to derive a novel iterative algorithm. Numerical simulation results show that our low-complexity algorithm provides near-optimal performance in terms of WSRSU while significantly outperforming conventional radio clustering and beamforming schemes. Additionally, the results also demonstrate the significant improvement in computing-resource utilization of Cloud-RAN over a traditional RAN with distributed computing resources.