Browne was born in Lanark in the Kingdom of Great Britain, which is presently in Scotland. Browne immigrated to the Thirteen Colonies with his family in 1754, and two years later, he became an officer in the British Royal Navy during the Seven Years' War. He sunk the Magnifique, a French sloop, off Sandy Bay, Lake Erie, in 1760, and retired in 1763 with the end of the war. He tended to a farm in Massachusetts until it was razed by the British Army in 1775 during the Lexington and Concord Campaign, upon which he took up a musket and volunteered as a soldier. He saved five soldiers trapped in a barn from redcoats, bayonetting the three British troops cornering them, and aided them to safety. For his services, he was named "Sergeant", and distinguished himself in the skirmishes around Boston, and when Boston fell in 1776, Browne led twenty militiamen to seize back his old brig, the HMS Aberdeen, and initiated a spree of anti-British piracy.
His death came on 15 April 1777 off the coast of Jamestowne, Virginia. His ship landed on the shore to drop off some excess cargo (including empty gunpowder barrels), and he oversaw the delivery. However, town sentries opened fire, and Browne was struck in the esophagus. Unable to breathe, he died of blood loss and suffocation.