This paper develops a non-asymptotic, local approach to quantitative propagation of chaos for a wide class of mean field diffusive dynamics. For a system of $n$ interacting particles, the relative entropy between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^2)$ at each time, as long as the same is true at time zero. A simple Gaussian example shows that this rate is optimal. The main assumption is that the limiting measure obeys a certain functional inequality, which is shown to encompass many potentially irregular but not too singular finite-range interactions, as well as some infinite-range interactions. This unifies the previously disparate cases of Lipschitz versus bounded measurable interactions, improving the best prior bounds of $O(k/n)$ which were deduced from global estimates involving all $n$ particles. We also cover a class of models for which qualitative propagation of chaos and even well-posedness of the McKean-Vlasov equation were previously unknown. At the center of a new approach is a differential inequality, derived from a form of the BBGKY hierarchy, which bounds the $k$-particle entropy in terms of the $(k+1)$-particle entropy.