William Vincent Wells was born in Boston in 1826, the son of poets Thomas Wells (1790–1861) and Anna Maria (Foster) Wells (1795–1868). He was a great-grandson of Massachusetts Governor Samuel Adams (1722–1803), and an uncle of mathematician Webster Wells, architect Joseph Morrill Wells, and silent film actress Mai Wells. William Wells grew up in Boston where he was educated "at the common schools of that city," but in 1834, his father left home to join the Navy, serving aboard the frigate USS Potomac as schoolmaster, and serving, from December 1836 until July 1838, aboard the USS Constitution as private secretary to Commodore Jesse Elliott, Commander of the Navy's Mediterranean Squadron. During his father's time abroad, Wells's mother earned her living as an educator (the historian Sidney Perley wrote in an 1889 sketch that "Mrs. Wells' chief attention was given to her school for young ladies").
