Richet was born in 1847 in Solesmes, a village in Normandy dominated in his childhood years by a growing Benedictine monastery. By the late 1860s he was a pupil of Narcisse-Virgile Diaz de la Peña (1807-1876). The Salon committee accepted Richet’s painting of the forest of Fontainebleau in 1869. By 1870, he had taken a studio in Paris at the southern edge of Montmartre.
Under the influence of Diaz, Richet depicted scenes around the village of Barbizon and in the Fontainebleau forest over the course of his career. Diaz died in 1876. After that, Richet worked under Jules Lefebvre, another painter of the Barbizon School. Lefebvre taught at the Académie Julian, which had been founded in 1868; it is likely that Richet enrolled there, and continued his study of landscapes. Richet enjoyed a long and settled career, painting both in the Barbizon area and in his native Normandy. He died in Paris in March of 1907 at age 60.