Born in 1890, he soon showed precocious drawing skills and entered the Birmingham School of Art at the age of twelve. A pupil at the Royal Academy Schools in 1907, he won the gold medal and a travelling scholarship in 1913, enabling him to visit both France and Italy. From 1915 to 1919 Brockhurst and his wife Anaïs lived in Ireland, where they were friendly with the artist Augustus John and his circle. Though he tried his hand at etching in 1914, it was not until 1920 that he began his career as an etcher in earnest, eventually achieving success as both a printmaker and society portraitist. Brockhurst held his first important exhibition in 1919, in London, and after it was well received returned to live there. In 1921 he was one of the early members of the newly-formed Society of Graphic Art and exhibited with them. Throughout the 1930s he continued an increasingly successful career as a portrait artist, with notable sitters including the film stars Merle Oberon and Marlene Dietrich, as well as the Duchess of Windsor, whose husband commissioned her portrait. In 1937 Brockhurst was elected to the Royal Academy and was able to command a price of 1,000 guineas for a portrait. In the same year however details of his relationship with his young model Kathleen Woodward, whom he had renamed Dorette, were made public after she gave an interview to the Sunday Express. In August 1939 Brockhurst and Dorette moved to the United States, and he was eventually divorced from his first wife in 1940.
In New York City, Brockhurst became both famous and rich with a series of society portraits but his printmaking output diminished, especially his etchings. He produced a few lithographs at the end of his career (around 1945). In 1951, he was elected into the National Academy of Design as an Associate member. Brockhurst died in 1978.