Gore was born in Boston in 1758, the tenth of thirteen children of Frances and John Gore, a successful merchant and artisan. He attended Boston Latin School and graduated from Harvard College in 1776, and served in the Continental Army as a clerk with an artillery regiment. After the war, he became a Boston lawyer and in 1785 married Rebecca Amory Payne, daughter of a wealthy merchant and maritime insurer.

Gore's political career began in 1788 when he was elected to represent Boston at the Philadelphia Convention to ratify the United States Constitution. He was also elected a member of the Massachusetts House of Representatives (1788-1789, and again in 1808). A year later President George Washington appointed Gore the first United States Attorney for Massachusetts, in which post he served 1789-1796.

In 1796, Washington appointed Gore as a commissioner to the Jay Treaty in Britain, in which post he served 1796-1803. Gore also spent two months as chargé d'affaires in London, 1803-1804, after his friend Rufus King resigned from his post. He remained abroad until 1804.

Soon after his return, Gore was elected to the Massachusetts Senate. He ran for Governor of the Commonwealth in 1807 and 1808 before winning a one-year term in 1809. He served as an overseer of Harvard University from 1810-1815 and later a fellow (1812-1820). In the spring of 1813 he was appointed to the U.S. Senate. He retired to his country home in Waltham, Massachusetts in 1816, where he remained until 1822 when declining health forced him to return to Boston. He died in 1827 in Waltham, and is buried in the Granary Burying Ground, Boston.