Empathy is critical to successful mental health support. Empathy measurement has predominantly occurred in synchronous, face-to-face settings, and may not translate to asynchronous, text-based contexts. Because millions of people use text-based platforms for mental health support, understanding empathy in these contexts is crucial. In this work, we present a computational approach to understanding how empathy is expressed in online mental health platforms. We develop a novel unifying theoretically-grounded framework for characterizing the communication of empathy in text-based conversations. We collect and share a corpus of 10k (post, response) pairs annotated using this empathy framework with supporting evidence for annotations (rationales). We develop a multi-task RoBERTa-based bi-encoder model for identifying empathy in conversations and extracting rationales underlying its predictions. Experiments demonstrate that our approach can effectively identify empathic conversations. We further apply this model to analyze 235k mental health interactions and show that users do not self-learn empathy over time, revealing opportunities for empathy training and feedback.