Livermore was born in Waltham in the Province of Massachusetts Bay, the son of Hannah (Brown) and Samuel Livermore,[1] and attended Waltham schools. He graduated from the College of New Jersey (now Princeton University) in 1752, then studied law, was admitted to the bar in 1756, and commenced practice in Waltham. He moved to Portsmouth, in 1758 and later to Londonderry. He was a member of the New Hampshire General Court (the state's general assembly) 1768–1769. He was judge-advocate in the Admiralty court and Attorney General from 1769 to 1774. He moved to Holderness in 1775 and was State attorney for three years.
Livermore was president of the State constitutional convention in 1791 and in 1792 was elected as a Federalist to the United States Senate and was reelected in 1798 and served from March 4, 1793, until his resignation effective June 12, 1801, due to ill health. He served as president pro tempore of the Senate during the Fourth and Sixth Congresses. The defunct town of Livermore, New Hampshire was named after him.