Spatial light interference microscopy (SLIM) is a powerful new quantitative phase optical imaging technique that can be used for studying live cells without the need for exogenous contrast agents. This paper proposes a novel deconvolution-based approach to reconstructing SLIM data, which dramatically improves the visual quality of the images. The proposed deconvolution formulation is tailored to the physics of SLIM imaging of biological samples, and a new fast algorithm is designed for computationally-efficient image reconstruction in this setting. Simulation and experimental results demonstrate that deconvolution can reduce the width of the point-spread function by at least 20%, and can significantly improve the contrast of high-resolution features. Temporally-resolved SLIM imaging with the high spatial resolution enabled by deconvolution provides new opportunities for studying the dynamics of cellular and sub-cellular processes.