Doxorubicin and Bortezomib in Treating Patients With Liver Cancer

Inclusion Criteria: * Patients must have microscopically confirmed hepatocellular carcinoma not amenable to curative resection; if patients have an isolated lesion in one lobe of the liver, a liver surgeon should determine resectability; central review is not required * Patients must have measurable disease as determined by Response Evaluation Criteria In Solid Tumors (RECIST) criteria, amenable to biopsy; patients are not mandated to allow biopsy, even though it is an important aspect of this clinical trial * Patients with history of malignancy treated within the past 5 years are not eligible; history of carcinoma-in-situ of cervix, squamous cell cancer of skin, basal cell cancer of skin, previously treated are allowed; others are excluded as recurrence of disease may confuse response rate and/or survival endpoints * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Patients must not have had prior systemic chemotherapy for HCC; patients on antineoplastics for non-malignant diseases, such as methotrexate for rheumatoid arthritis, are allowed, providing patients have been off these agents for at least 4 weeks and all related toxicities have resolved to baseline * Patients may have had prior embolization without chemotherapy; patients who have had chemoembolization are not eligible; patients may have had radiofrequency (RF) ablation, cryosurgery or ethanol injection; patients must have documented progression with the involved lesion or at least one previously untreated lesion amenable to biopsy * Platelet count must be \>= 100,000/mm\^3 in absence of splenomegaly; platelet count must be \>= 75,000/mm\^3 with splenomegaly * Absolute neutrophil count (ANC) must be \>= 1,500/mm\^3 in absence of splenomegaly; ANC must be =\< 1,000/mm\^3 with splenomegaly * Alkaline phosphate (ALT) must be =\< 5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) must be =\< 5 x institutional ULN * Bilirubin must be =\< 2 mg/dl * Patients may not exhibit Child Pugh scale grade C cirrhosis * Serum creatinine=\< 2.0 mg/dl * All females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy * Women of childbearing potential and sexually active males are strongly advised to use an accepted and effective method of contraception * Patients must not have known bleeding diathesis, international normalized ratio (INR) \> 1.5 or Partial thromboplastin time (PTT) \> 1.5 x institutional ULN (required due to biopsy portion of study); use of vitamin K or fresh frozen plasma to correct values just prior to biopsy or enrollment is not allowed are not eligible Exclusion criteria: * Patients have baseline peripheral neuropathy \> grade 1 * Patients with history of untreated malignancy other than HCC * Patients have had prior use of octreotide or tamoxifen as therapy for HCC * Patients with known allergy to boron, mannitol or bortezomib * Women are pregnant or breast-feeding (due to the uncertain effects of bortezomib in the developing fetus and young infants) * Patients have an underlying medical condition that precludes safe participation in this clinical trial * Patients have psychiatric illness or continued substance abuse that may impair the ability to provide informed consent or prevent safe administration of bortezomib * Patients with ejection fraction (EF) \< 50% measured by Echocardiography (ECHO) or Multiple gated acquisition (MUGA) * Patients on verapamil who cannot be switched to an alternative medication (due to the interaction with doxorubicin)