CD8+ Antigen-Specific T Cells, Cyclophosphamide, Aldesleukin, and Ipilimumab in Treating Patients With Metastatic Melanoma

Inclusion Criteria: * ELIGIBILITY FOR ENROLLMENT * Histopathologic documentation of melanoma concurrent with the diagnosis of metastatic disease * Expression of human leukocyte antigen (HLA)-A2 * Eastern Cooperative Oncology Group (ECOG)/Zubrod performance status of '0-1' at screening visit * Women of childbearing potential (WOCBP) must be using an adequate method of contraception to avoid pregnancy throughout the study in such a manner that the risk of pregnancy is minimized; suggested precautions should be used to minimize the risk of pregnancy for at least 1 month before start of therapy, and while women are on study for up to 3 months after T cell infusion, and at least 8 weeks after the study drug is stopped; WOCBP include any female who has experienced menarche and who has not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation or bilateral oophorectomy) or is not postmenopausal * Men must be willing and able to use an acceptable method of birth control, for at least 3 months after completion of the study, if their sexual partners are WOCBP * Willing and able to give informed consent * Adequate venous access - consider peripherally inserted central catheter (PICC) or central line * Evaluation of v-raf murine sarcoma viral oncogene homolog B (BRAF)V600 mutation status * Measurable tumor (by Response Evaluation Criteria in Solid Tumors \[RECIST\] criteria) * Melan-A (MART) 1 or solute carrier family 45, member 2 (SLC45A2) (+) staining results; (if patients have not had staining test in the past, the test will be run after patient consent is obtained, but before enrollment) * ELIGIBILITY FOR TREATMENT (INCLUDES CYCLOPHOSPHAMIDE, T CELL, ANTI-CTLA4 INFUSIONS AND SC IL-2) * ECOG/Zubrod performance status of '0-1' * At least 4 weeks must have elapsed since the last chemotherapy, radiotherapy or major surgery; at least 6 weeks for nitrosoureas, mitomycin C and liposomal doxorubicin; if started before T-cell administration, ipilimumab infusions must be least 21 days apart * Toxicity related to prior therapy must either have returned to =\< grade 1, baseline, or been deemed irreversible * Persons of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 8 weeks after study drug is stopped * Willing and able to give informed consent. Exclusion Criteria: * EXCLUSION FOR ENROLLMENT * Any other malignancy from which the patient has been disease-free for less than 5 years, with the exception of adequately treated and cured basal or squamous cell skin cancer, superficial bladder cancer, carcinoma in situ of the cervix * Pregnant women, nursing mothers, men or women of reproductive ability who are unwilling to use effective contraception; women of childbearing potential with a positive pregnancy test within 3 days prior to entry * Active and untreated central nervous system (CNS) metastasis (including metastasis identified during screening magnetic resonance imaging \[MRI\] or contrast computed tomography \[CT\]) * No signs or symptoms of CNS metastases (mets) within the last 30 days (from enrollment evaluation) * No single lesion larger than 1 cm * No more than 5 lesions * Autoimmune disease: patients with a history of inflammatory bowel disease are excluded from this study, as are patients with a history of autoimmune disease (e.g. systemic lupus erythematosus, vasculitis, infiltrating lung disease) whose possible progression during treatment would be considered by the investigator to be unacceptable * Any underlying medical or psychiatric condition, which in the opinion of the investigator, will make the administration of study drug hazardous or obscure the interpretation of adverse events, such as a condition associated with frequent diarrhea * Positive screening tests for human immunodeficiency virus (HIV), hepatitis B (hep B), and hepatitis C (hep C) (referencing blood draw at leukapheresis screening); if positive results are not indicative of true active or chronic infection, the patient can be treated * White blood cells (WBC) =\< 1000/uL * Hematocrit (Hct) =\< 24% or hemoglobin (Hb) =\< 8 g/dL * Absolute neutrophil count (ANC) =\< 500 * Platelets =\< 50,000 * Creatinine \>= 3.0 x upper limit of normal (ULN) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \>= 2.5 x ULN * Bilirubin \>= 3 x ULN * Steroids are not permitted 3 days prior to T cell infusion and concurrently during therapy * Any non-oncology vaccine therapy used for the prevention of infectious disease within 1 month before or after any ipilimumab dose * Patients may not be on any other treatments for their cancer aside from those included in the protocol; patients may not undergo another form of treatment concurrently with this study * EXCLUSION CRITERIA FOR TREATMENT * WBC =\< 1000/uL (prior to cyclophosphamide and T cell infusions) * Hct =\< 24% or hemoglobin =\< 8 g/dL (prior to cyclophosphamide and T cell infusions) * ANC =\< 500 (prior to cyclophosphamide and T cell infusions) * Platelets =\< 50,000 (prior to cyclophosphamide and T cell infusions) * Creatinine \>= 3.0 x ULN (prior to cyclophosphamide and T cell infusions) * AST/ALT \>= 2.5 x ULN (prior to cyclophosphamide and T cell infusions) * Bilirubin \>= 3 x ULN (prior to cyclophosphamide and T cell infusions) * Pregnant women, nursing mothers, men or women of reproductive ability who are unwilling to use effective contraception; women of childbearing potential with a positive pregnancy test within 3 days prior to entry. * Steroids are not permitted 3 days prior to T cell infusion and concurrently during therapy. * Any non-oncology vaccine therapy used for the prevention of infectious disease within 1 month before or after any ipilimumab dose. * Patients may not be on any other treatments for their cancer aside from those included in the protocol. Patients may not undergo another form of treatment concurrently with this study. * Active and untreated central nervous system (CNS) metastasis (including metastasis identified during screening MRI or contrast CT): * No signs or symptoms of CNS mets within the last 30 days (from enrollment evaluation). * No single lesion larger than 1cm * No more than 5 lesions