He was born at Saint-Cloud, near Paris, as the only child of Augustin Filon, the French littérateur who was appointed as the official tutor to the Prince Imperial.
Accompanying the Prince Imperial in his exile, the Filon family came to England in 1878 and lived at Margate.
In 1894 Filon became a student at University College London and received his BA in 1896 with a gold medal in Greek.
In 1912, upon Pearson's retirement, Filon was appointed to the Goldsmid Chair of Applied Mathematics and Mechanics at University College, London.
He was a member of the University of London Senate (1920) and the court, dean of the Faculty of Science, chairman of the Academic Council (1924–33) and vice-chancellor (1933–35).