A novel dynamic radio-cooperation strategy is proposed for Cloud Radio Access Networks (C-RANs) consisting of multiple Remote Radio Heads (RRHs) connected to a central Virtual Base Station (VBS) pool. In particular, the key capabilities of C-RANs 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 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. Our approach aims for a suboptimal solution by transforming the original problem into a Mixed-Integer Second-Order Cone Program (MI-SOCP), which can be solved efficiently using a proposed iterative algorithm. Numerical simulation results show that our low-complexity algorithm provides close-to-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 C-RANs over traditional RANs with distributed computing resources.