This paper investigates the compress-and-forward scheme for an uplink cloud radio access network (C-RAN) model, where multi-antenna base-stations (BSs) are connected to a cloud-computing based central processor (CP) via capacity-limited fronthaul links. The BSs compress the received signals with Wyner-Ziv coding and send the representation bits to the CP; the CP performs the decoding of all the users' messages. Under this setup, this paper makes progress toward the optimal structure of the fronthaul compression and CP decoding strategies for the compress-and-forward scheme in C-RAN. On the CP decoding strategy design, this paper shows that under a sum fronthaul capacity constraint, a generalized successive decoding strategy of the quantization and user message codewords that allows arbitrary interleaved order at the CP achieves the same rate region as the optimal joint decoding. Further, it is shown that a practical strategy of successively decoding the quantization codewords first, then the user messages, achieves the same maximum sum rate as joint decoding under individual fronthaul constraints. On the joint optimization of user transmission and BS quantization strategies, this paper shows that if the input distributions are assumed to be Gaussian, then under joint decoding, the optimal quantization scheme for maximizing the achievable rate region is Gaussian. Moreover, Gaussian input and Gaussian quantization with joint decoding achieve to within a constant gap of the capacity region of the Gaussian multiple-input multiple-output (MIMO) uplink C-RAN model. Finally, this paper addresses the computational aspect of optimizing uplink MIMO C-RAN by showing that under fixed Gaussian input, the sum rate maximization problem over the Gaussian quantization noise covariance matrices can be formulated as convex optimization problems, thereby facilitating its efficient solution.