He was born the son of the writer and judge Hugh Henry Brackenridge in Pittsburgh, Pennsylvania on May 11, 1786. Educated by his father and private tutors, he attended a French academy at St. Genevieve, Louisiana (now Missouri). He studied law and was admitted to the Pennsylvania bar in 1806, then practiced in Somerset, Pennsylvania.
He moved to St. Louis, Missouri, where he was a lawyer and journalist. In 1811 Brackenridge was the first recorded tourist to present-day South Dakota, hosted by fur trader Manuel Lisa. Henry was appointed deputy attorney general of the Territory of Orleans (Louisiana), and district judge of Louisiana in 1812.
He played an intelligence role during the War of 1812, and in 1814 published a history of the war. In 1817 he was appointed secretary of a mission to South America. He was elected a member of the American Antiquarian Society in 1818. Brackenridge in 1821 entered the diplomatic service of General Andrew Jackson, who was the new commissioner of Florida. Through Jackson's influence, he served as U.S. judge for the western district of Florida 1821–32.
Brackenridge is a borough in Allegheny County in the U.S. state of Pennsylvania, along the Allegheny River. It is part of the Pittsburgh Metro Area.