Weeks was born in Boston, Massachusetts in 1849. His parents were affluent spice and tea merchants from Newton, a suburb of Boston, and as such they were able to finance their sons youthful interest in painting and travelling. As a young man Weeks visited the Florida Keys to draw, and also travelled to Surinam in South America. His earliest known paintings date from 1867 when he was eighteen years old, although it is not until his Landscape with Blue Heron, dated 1871 and painted in the Everglades, that Weeks started to exhibit a dexterity of technique and eye for composition—presumably having taken professional tuition.
In 1872 Weeks relocated to Paris, becoming a pupil of Leon Bonnat and Jean-Leon Gerome.