This pilot phase I trial studies the side effects and best dose of genetically modified T-cells followed by aldesleukin in treating patients with stage III-IV melanoma. T-cells are a type of white blood cell that help the body fight infections. Genes that may help the T-cells recognize melanoma cells are placed into the T-cells in the laboratory. Adding these genes to the T cells may help them kill more tumor cells when they are put back in the body. Aldesleukin may enhance this effect by stimulating white blood cells to kill more melanoma cells.