Radio Access Network (RAN) slicing is a promising technique to realize service multiplexing in the upcoming Beyond Fifth-Generation (B5G) and Sixth-Generation (6G) networks. However, limited resources and substantial interference pose an obstacle to efficient RAN slicing. In this paper, a novel Reconfigurable Intelligent Surface (RIS) assisted Cloud RAN (C-RAN) slicing framework is proposed, where enhanced Mobile BroadBand (eMBB) and Ultra-Reliable Low-Latency Communications (URLLC) services coexist via Non-Orthogonal Multiple Access (NOMA) technology. To guarantee the high system performance in considered service multiplexing scenarios, a slicing orchestration optimization problem is formulated for the joint optimization of user clustering, beamforming design, reflection coefficients design, and power allocation for each NOMA cluster. To solve the obtained non-convex problem, an Alternating Optimization (AO) based iterative algorithm is developed. Specifically, an improved K-means-based clustering algorithm is first proposed to design the NOMA clusters. Then, the problem is decoupled into three sub-problems, which are solved alternatingly by applying Fractional Programming (FP) and Successive Convex Approximation (SCA). It is proved that the proposed algorithm can converge with polynomial computational complexity. Numerical results show that the proposed algorithm can significantly improve the performance of the slicing system compared to the benchmark schemes.