4th of Dalswinton. Son of Hugh Maxwell, 3rd of Dalswinton and Jane (or Jean) Douglas, a daughter of Helen Erskine, Boswell's mother's half-sister, and William Douglas, 2nd of Kelhead. Married to Mary Boscawen, a daughter of the Viscount Falmouth. Brother of George Maxwell. Due to financial troubles, he had to sell Dalswinton in 1786.
Maxwell eventually achieved the rank of Major in the army, so he might have been a Captain in 1763, although Boswell's use of the title "Captain" is not always to be taken literally.