Systemic Therapy With Interferon, Interleukin-2 and BRAF Inhibitor

Study Groups: If you are found to be eligible to take part in this study, you will be assigned to a study group based on when you join this study. Up to 4 groups of 6 participants will be enrolled in the Phase I portion of the study and up to 53 participants will be enrolled in Phase II. If you are enrolled in the Phase I portion, the dose of vemurafenib and aldesleukin you receive will depend on when you joined this study. The first group of participants will receive the lowest dose levels of vemurafenib and aldesleukin. Each new group will receive a higher dose of vemurafenib or aldesleukin than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable doses of vemurafenib and aldesleukin are found. If you are enrolled in the Phase II portion, you will receive vemurafenib and aldesleukin at the highest doses that were tolerated in the Phase I portion. All participants will receive the same dose level interferon alfa-2b. Study Drug Administration: Each study cycle is 21 days. You will take vemurafenib by mouth twice a day (1 time in the morning and 1 time in the evening) every day of each study cycle. Vemurafenib can be taken with or without food. If you miss a scheduled dose of vemurafenib, do not try to "make up" the dose by taking 2 doses at the next scheduled dosing time. Continue taking the study drug as scheduled. You will receive aldesleukin by vein over 96 hours on Days 1-4 of each study cycle. To receive aldesleukin, you will have a central venous catheter (CVC) inserted, if you do not already have one. A CVC is a sterile flexible tube that will be place into a large vein while you are under local anesthesia. Your doctor will explain this procedure to you in more detail, and you will be required to sign a separate consent form. You will receive interferon alfa-2b as an injection under your skin on Days 1-5 of each study cycle. Supportive Drugs: You will be given other drugs to help lower the risk of side effects. The study staff will tell you about these drugs, how they will be given, and the possible risks. Study Visits: The following tests and procedures will be performed: On Day 1 (+/- 3 days) of each cycle: * Your performance status will be recorded. * Your vital signs will be measured. * You will be asked about any drugs you may be taking and any side effects you may have had. * If you are able to become pregnant, you will have a blood (about 1 teaspoon) or urine pregnancy test. * You will have an EKG. Your skin will be checked for non-melanoma skin cancers on Day 1 of Cycle 1 (+/- 3 days) and then every 3 cycles. Blood (about 1 teaspoon) will be drawn for routine tests on Days 1, 8, and 15 of each cycle. On Day 1 of each cycle, some of this blood will be used to check your liver and kidney function. At the end of each cycle (Day 21), you will have a physical exam, including measurement of your weight. Any tumor that can be felt with the hands will be measured with measuring tape during the physical exam to see if it has changed size. On Day 21 (+/- 7 days) of every 2 cycles, you will have a chest x-ray and a CT or MRI scan to check the status of the disease. Anytime the doctor thinks it is needed, photos of any skin lesions you have will be taken. Your private areas will be covered (as much as possible), and a picture of your face will not be taken unless there are lesions on your face. Length of Study: You may take the study drug combination for as long as the doctor thinks you are benefitting. You will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if you are unable to follow study directions. Your participation on the study will be over once you have completed the end-of-treatment visit and follow-up. End-of-Treatment Visit Within 14 days after you stop taking the study drug combination, the following tests and procedures will be performed: * Your performance status will be recorded. * You will have a physical exam, including measurement of your vital signs and weight. * You will be asked about any drugs you may be taking and any side effects you may have had. vBlood (about 2 tablespoons) will be drawn for routine tests. ° If the doctor thinks it is needed, you will have a CT or MRI scan. Follow-Up Every 3 months for up to 3 years, you will be contacted by phone or during a clinic visit to see how you are doing. The phone call should last about 5 minutes. This is an investigational study. Vemurafenib, interferon alfa-2b, and Aldesleukin are FDA approved and commercially available to treat metastatic cancer. Giving these drugs in combination is investigational. The study doctor will tell you how the study drugs are designed to work. Up to 71 patients will take part in this study. All will be enrolled at MD Anderson.