We consider the task of automated estimation of facial expression intensity. This involves estimation of multiple output variables (facial action units — AUs) that are structurally dependent. Their structure arises from statistically induced co-occurrence patterns of AU intensity levels. Modeling this structure is critical for improving the estimation performance; however, this performance is bounded by the quality of the input features extracted from face images. The goal of this paper is to model these structures and estimate complex feature representations simultaneously by combining conditional random field (CRF) encoded AU dependencies with deep learning. To this end, we propose a novel Copula CNN deep learning approach for modeling multivariate ordinal variables. Our model accounts for $ordinal$ structure in output variables and their $non$-$linear$ dependencies via copula functions modeled as cliques of a CRF. These are jointly optimized with deep CNN feature encoding layers using a newly introduced balanced batch iterative training algorithm. We demonstrate the effectiveness of our approach on the task of AU intensity estimation on two benchmark datasets. We show that joint learning of the deep features and the target output structure results in significant performance gains compared to existing deep structured models for analysis of facial expressions.