The purpose of this study is to see if the combination of entinostat and pembrolizumab can be an effective treatment for patients with melanoma of the eye (uveal melanoma) that has spread to other sites of the body (metastatic disease). Pembrolizumab is an antibody that helps the immune system to attack cancer cells. Although pembrolizumab has proven clinical efficacy in treating patients with metastatic cutaneous melanoma, an effect on metastatic uveal melanoma has not been established. Entinostat is a histone deacetylase (HDAC) inhibitor that has effects on both cancer cells and immune regulatory cells, thus potentially enhancing the effects of immunotherapy.