Greene was born on December 26, 1901, in Washington, D.C. Shortly after graduating high school, he went to live with his aunt. He was influenced to study medicine by two of his uncles who were both physicians.
Greene first had plans to attend dental school. He attended the University of Pennsylvania for two years and then continued into the school’s dental program. He graduated with a D.D.S. degree in 1926 but realized dentistry was not his passion. He enrolled at Harvard University where he spent 2 years in the premed training program from 1927 to 1929. In 1930, Greene interned in Cleveland City Hospital. Then he returned to the University of Pennsylvania and earned a Bachelor of Arts degree (1932). He received his M.D. at the College of Medicine at Howard University in 1936.
Greene did a residency in Douglass Hospital in Philadelphia under Dr. John P. Turner. Then he moved to Freedmen’s Hospital in Washington D.C. for two years. Greene served as a diplomat of the American Board of Surgery in 1943. That same year he was appointed an assistant professor of surgery at Howard.