Maxwell was born in Dalbeattie, Kirkcudbrightshire. He attended Edinburgh College of Art from 1921 to 1926 and then travelled in Europe for a year. For a short period he, like several other Scottish artists of his generation, studied in Paris, where he experimented with a cubist style before settling on a more conventional style of painting. In 1929 Maxwell began teaching at Edinburgh College of Art, where he became a colleague of William Gillies, who often accompanied him on painting holidays. In the 1950s Maxwell painted an extended series of semi-abstract still lifes, reminiscent of the work of Braque.