Clinical subtyping, a critical component of personalized medicine, classifies patients with a particular disease into distinct subgroups based on their unique features. However, conventional data-driven subtyping approaches often entail a manual characterization of the identified clusters, complicating the task due to the high dimensionality and heterogeneity of the data. In this work, we propose a novel framework for interpretable clinical subtyping using deep metric learning. Our proposed pipeline unifies prior approaches to clinical subtyping, and introduces automatic characterization of the learned clusters in an interpretable and clinically meaningful manner. We showcase the effectiveness of this framework on real-world clinical case studies, demonstrating its utility in uncovering actionable clinical knowledge.