This paper studies the downlink of a cloud radio access network (C-RAN) in which a centralized processor (CP) communicates with mobile users through base stations (BSs) that are connected to the CP via finite-capacity fronthaul links. Information theoretically, the downlink of a C-RAN is modeled as a two-hop broadcast-relay network. Among the various transmission and relaying strategies for such model, this paper focuses on the compression strategy, in which the CP centrally encodes the signals to be broadcasted jointly by the BSs, then compresses and sends these signals to the BSs through the fronthaul links. This paper characterizes an achievable rate region for a generalized compression strategy with Marton's multicoding for broadcasting and multivariate compression for fronthaul transmission. We then compare this rate region with the distributed decode-forward (DDF) scheme, which achieves the capacity of the general relay networks to within a constant gap, and show that the difference lies in that DDF performs Marton's multicoding and multivariate compression jointly as opposed to successively as in the compression strategy. A main result of this paper is that under the assumption that the fronthaul links are subject to a \emph{sum} capacity constraint this difference is immaterial, so the successive encoding based compression strategy can already achieve the capacity region of the C-RAN to within a constant gap, where the gap is independent of the channel parameters and the power constraints at the BSs. For the special case of the Gaussian network, we further establish that under individual fronthaul constraints, the compression strategy achieves to within a constant gap to the \emph{sum} capacity of the C-RAN.