This phase I trial is studying the side effects and best dose of recombinant interferon alfa-2b when given together with azacitidine in treating patients with stage III or stage IV melanoma or stage IV kidney cancer that cannot be removed by surgery. Drugs used in chemotherapy, such as azacitidine, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. Recombinant interferon alfa-2b may interfere with the growth of tumor cells. Giving azacitidine together with recombinant interferon alfa-2b may kill more tumor cells.