There is a long history of establishing central limit theorems for Markov chains. Quantitative bounds for chains with a spectral gap were proved by Mann and refined later. Recently, rates of convergence for the total variation distance were obtained for random walks on expander graphs, which are often used to generate sequences satisfying desirable pseudorandom properties. We prove a local central limit theorem with an explicit rate of convergence for random walks on expander graphs, and derive an improved bound for the total variation distance.