Studied law at Trinity Hall, Cambridge, to which he was admitted on Nov 1, 1757. He became a lawyer at Lincoln's Inn, London, and a fellow of the Society of Antiquaries.1 In the early 1800s Claxton had built Shirley House, near Croydon, Surrey, which was designed by architect John Shaw (1776-1832).2 The site is now home to the Trinity School.
Claxton was, at some time, a close friend of William Temple.
Claxton died at Bath on March 4, 1811.3
Boswell was introduced to Claxton by Temple, who knew him from Cambridge, on May 15, 1763. Apparently, they took a liking to each other and met occasionally whenever Boswell was in London for the years to come. At one point, Boswell even borrowed a wig and gown from Claxton.