Allen was born in 1770, the eldest son in the Quaker family of Job Allen (1734–1800), a silk manufacturer, and his wife Margaret Stafford (died 1830). He was educated at a Quaker school in Rochester, Kent, and then went into his father's business. As a young man in the 1790s, he became interested in science. He attended meetings of scientific societies, including lectures at St. Thomas's Hospital and Guy's Hospital, becoming a member of the Chemical Society of the latter establishment. On Job Allen's death, the family silk firm was taken over by his father's assistant.