Diagnostic interpretation of breast MRI studies requires meticulous work and a high level of expertise. Computerized algorithms can assist radiologists by automatically characterizing the detected lesions. Deep learning approaches have shown promising results in natural image classification, but their applicability to medical imaging is limited by the shortage of large annotated training sets. In this work, we address automatic classification of breast MRI lesions using two different deep learning approaches. We propose a novel image representation for dynamic contrast enhanced (DCE) breast MRI lesions, which combines the morphological and kinetics information in a single multi-channel image. We compare two classification approaches for discriminating between benign and malignant lesions: training a designated convolutional neural network and using a pre-trained deep network to extract features for a shallow classifier. The domain-specific trained network provided higher classification accuracy, compared to the pre-trained model, with an area under the ROC curve of 0.91 versus 0.81, and an accuracy of 0.83 versus 0.71. Similar accuracy was achieved in classifying benign lesions, malignant lesions, and normal tissue images. The trained network was able to improve accuracy by using the multi-channel image representation, and was more robust to reductions in the size of the training set. A small-size convolutional neural network can learn to accurately classify findings in medical images using only a few hundred images from a few dozen patients. With sufficient data augmentation, such a network can be trained to outperform a pre-trained out-of-domain classifier. Developing domain-specific deep-learning models for medical imaging can facilitate technological advancements in computer-aided diagnosis.