 Archibald was baptized 24 August 1739 at Inveraray, Scotland.He was the second son of James Campbell (1706–1760) 3rd of Tuerechan (8th Chief of Tearlach, descended from Clan Campbell of Craignish), Commissary of the Western Isles of Scotland, and Elizabeth (died 1790), daughter of James Fisher, Provost of Inveraray.He grew up with his family at Dunderave Castle, and enjoyed the patronage of both Archibald Campbell, 3rd Duke of Argyll and Henry Dundas, 1st Viscount Melville.