Born in Liverpool, England to West Indian immigrants, he graduated from Eton College in 1859, and served in the British Royal Navy. He moved to South Carolina in 1867 and established a law practice. Elliott helped organize the local Republican Party and served in the state constitutional convention. In 1868 he was elected to the South Carolina House of Representatives. The next year he was appointed assistant adjutant-general; he was the first African American commanding general of the
Elliott was elected as a Republican to the Forty-second and Forty-third United States Congress. He "delivered a celebrated speech" in favor of the Civil Rights Act of 1875 He resigned on November 1, 1874, to fight political corruption in South Carolina. He served again in the South Carolina House of Representatives, where he was elected as Speaker of the House.