In this paper, we address fish species identification in underwater video for marine monitoring applications such as the study of marine biodiversity. Video is the least disruptive monitoring method for fish but requires efficient techniques of image processing and analysis to overcome challenging underwater environments. We propose two Deep Convolutional Neural Network (CNN) approaches for fish species classification in unconstrained underwater environment. In the first approach, we use a traditional transfer learning framework and we investigate a new technique based on training/validation loss curves for targeted data augmentation. In the second approach, we propose a hierarchical CNN classification to classify fish first into family levels and then into species categories. To demonstrate the effectiveness of the proposed approaches, experiments are carried out on two benchmark datasets for automatic fish identification in unconstrained underwater environment. The proposed approaches yield accuracies of 99.86% and 81.53% on the Fish Recognition Ground-Truth dataset and LifeClef 2015 Fish dataset, respectively.