Greene was born in Providence, Rhode Island, on June 27, 1850. He attended the United States Military Academy at West Point and graduated in 1870. He first served in the U.S. artillery and then transferred to the Corps of Engineers in 1872. He next served as an attaché from the War Department to the U.S. legation in St. Petersburg, Russia. While there he served in the Russian army during its war with Turkey. He was promoted to first lieutenant in 1874 and captain in 1883. He returned to the U.S. and was a civil engineer to the city of Washington, D.C. and was a professor of artillery at West Point before resigning from the Army on December 31, 1886. He published multiple articles on the development of modern warfare and U.S. military policy.