He attended a French Academy at Montreal, Canada, studied law, and was admitted to the bar in 1790.
He practiced law in Poughkeepsie and served as the county clerk of Herkimer County, New York, from 1791 to 1798. He was also the county clerk of Oneida County, New York, from 1798 to 1802. Platt was elected as a Federalist to the Sixth Congress, and served from March 4, 1799, to March 3, 1801.
He was the chairman of the United States House Committee on Revisal and Unfinished Business.
Afterward, he resumed the practice of law and served as a general in the Cavalry in the New York State Militia. In 1810, he was an unsuccessful candidate for governor.
From 1814 to 1821, Platt was an associate justice of the New York Supreme Court. He was a delegate to the New York Constitutional Convention in 1821.
He was buried at the Riverside Cemetery in Plattsburgh.
He was a member of the United States House of Representatives. He was a member of the New York State Assembly in 1796. He was a member of the New York State Senate from 1810 to 1813.
He was also a member of the Council of Appointment in 1813.