Grove was born in Fayetteville, North Carolina in 1764. After studying law, he was admitted to the state bar and became a practicing attorney. In 1786, Grove was first elected to the North Carolina House of Commons; he would serve again in 1788 and 1789. In 1788, Grove was a delegate to the state convention that considered ratification of the United States Constitution; he voted against postponement of ratification, but the state did not ratify the constitution at that time. Grove was also a delegate to the 1789 state convention where North Carolina finally ratified the federal constitution.
A trustee of the University of North Carolina at Chapel Hill and the president of the Fayetteville Branch of the Bank of the United States, Grove was elected to the Second United States Congress in 1790; he was re-elected to the 3rd through 7th Congresses as a Federalist, serving consecutively from March 4, 1791 to March 3, 1803. Although he ran for re-election in 1802, he was defeated for a seat in the 8th U.S. Congress.
Grove died in 1818 and is buried in Fayetteville's Grove Creek Cemetery.