The son of a clergyman, he was educated in Dublin and London and held a variety of clerical posts in the government from 1756 to 1773. Francis may have written the Letters of Junius, a series of bitter lampoons against the government of King George III published by a London newspaper from 1769 to 1772, when he was a clerk in the war office.

In June 1773 the prime minister, Lord Frederick North, appointed him a member of the newly created four-man council that was to rule British possessions in India with Governor-General Hastings (Hastings, Warren). Francis led two of his colleagues in a struggle against Hastings; in part because he coveted Hastings' job, but there were also differences between the two men on policy matters, including land-revenue collection. Although Hastings gained the upper hand by 1776—after two of the opposing councillors had died—Francis continued his attacks, and in 1780 the governor-general wounded him in a duel. Returning to England in 1781, Francis turned public opinion against Hastings with a series of anonymous pamphlets. He entered Parliament in 1784 and was the moving spirit behind Hastings' impeachment, begun in 1788. The acquittal of Hastings in 1795 embittered Francis deeply and led to his defeat in a parliamentary election. He served again in Parliament from 1802 to 1807, when he retired from politics. He was knighted in 1806.