Selumetinib and Akt Inhibitor MK2206 in Treating Patients With Stage III or Stage IV Melanoma Who Failed Prior Therapy With Vemurafenib or Dabrafenib

Inclusion Criteria:

* Patients must have incurable unresectable stage III or IV histologically confirmed Melanoma with V600-mutant BRAF disease and must have progressed after therapy on selective BRAF inhibitor; all patients must have biopsiable tumor and a biopsy must be performed with the collection of FFPE and if possible FF prior to initiation of treatment on this protocol; archival tumor tissue must also be obtained if at all available; this required biopsy will not be necessary if a previous biopsy of progressing tumor after selective BRAF therapy had already been obtained and is adequate
* Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional techniques or as ≥ 10 mm with spiral computed tomography (CT) scan
* Patients must have received prior therapy and progressed following a selective BRAF inhibitor (i.e., vemurafenib, dabrafenib, LGX818, etc.); patients must have completed prior therapy a minimum of 4 weeks previously (6 weeks for BCNU and/or mitomycin C), 4 weeks for prior biologic therapy, and 2 weeks for localized radiation therapy; all treatment related toxicity must have resolved to grade 2 or less as well; patients may initiate the protocol treatment at 48 hours following the completion of BRAF inhibitor; patients must have had no more than 2 prior chemotherapy regimens; patients cannot receive chemotherapy after the BRAF inhibitor treatment and prior to enrollment on this protocol; up to two prior immunotherapy regimens for advanced disease are allowed and one may be given between BRAF inhibitor therapy and this trial
* Patients must not be refractory to the BRAF inhibitor; patients must demonstrate some degree of tumor regression initially on BRAF inhibitor prior to progression; (tumor regression does not require RECIST objective response); they cannot have progressive disease at the time of first evaluation (4 or 8 weeks) on the BRAF inhibitor
* Baseline Ophthalmologic exam must be done at screening to include slit lamp exam and fundoscopy; an OCT scan should be considered in case of retinal abnormality at exam
* Life expectancy of greater than or equal to 3 months
* Eastern Cooperative Oncology Group (ECOG) performance status ≤1 (Karnofsky ≥ 70%)
* Absolute neutrophil count ≥ 1,500 mm³
* Hemoglobin ≥ 9.0 g/dL (patients may be transfused to achieve level)
* Platelet count ≥ 100,000/μL
* Serum glutamic oxaloacetic transaminase (SGOT) and serum glutamic pyruvate transaminase(SGPT) \< 2.5 X upper limit of normal (ULN)
* Total bilirubin \< 1.5 mg/dL
* Serum creatinine ≤ 2.0 mg/dL OR creatinine clearance \> 50 mL/min, determined by 24-hour urine collection
* Fasting blood glucose \< 160 mg/dL OR
* HgbA1C \< 8% disease (uncontrolled diabetes)
* Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation
* Patients must have a negative serum pregnancy test prior to being eligible to take part in the study
* Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with AZD6244 hydrogen sulfate and MK-2206
* Baseline echocardiogram or MUGA must be performed at screening and patients must have LVEF \> 55%; additionally baseline EKG must be performed and corrected QTc must be \< 480 milliseconds
* Baseline electrocardiogram(EKG) must be performed and corrected QTc must be \< 480 milliseconds
* Patients must be able to swallow tablets and capsules to participate in the study
* No history of allergic reactions attributed to compounds of similar chemical or biologic composition to AZD6244 hydrogen sulfate, MK-2206, or other agents used in the study
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, psychiatric illness/social situations that would limit compliance with study requirements, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, uncontrolled hypertension (BP \>= 150/95 despite optimal therapy), baseline ejection fraction \< 55% or the lower limit of institutional normal, heart failure NYHA Class II or above, prior or current cardiomyopathy, atrial fibrillation with heart rate \> 100 bpm, and uncontrolled angina (Canadian Cardiovascular society grade II-IV despite medical therapy); acute coronary syndrome within 6 months from starting therapy
* Patients must have completed prior therapy a minimum of 4 weeks previously (6 weeks for BCNU and/or mitomycin C), 4 weeks for prior biologic therapy, and 2 weeks for localized radiation therapy
* All treatment-related toxicity must have resolved to grade 2 or less
* No patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier
* Patients must have had no more than 2 prior chemotherapy regimens
* Patients cannot receive chemotherapy after the BRAF-inhibitor treatment and prior to enrollment on this protocol
* Up to two prior immunotherapy regimens for advanced disease are allowed and one may be given between BRAF-inhibitor therapy and this trial
* Patients may not be receiving any other investigational agents at the same time as study treatment
* Patients receiving medications or substances that are strong inhibitors or inducers of cytochrome P450 3A4 (CYP3A4) are ineligible
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Patients must not have received chemotherapy in the time between the failure of BRAF inhibitor and the enrollment onto the present trial