Alexander McDonnell was born in Belfast, Ireland, the son of a famous surgeon (also Alexander).
Alexander left Belfast in 1816 to work in Demarara in the West Indies. He became a noted West Indian merchant.
In 1820 he became Secretary of the Committee of West Indian Merchants, where his duties were to watch the progress through Parliament of bills connected with the West Indies.
In 1825 he became a pupil of William Lewis, a leading London chess player.
In 1831 he joined the Westminster Chess Club, founded by George Walker, where he became the strongest London player.
In June 1834, the Westminster Chess Club issued a challenge on McDonnell's behalf, and a match was prepared between McDonnell and La Bourdonnais. McDonnell won 27, drew 13, and lost 45 games.
McDonnell developed a kidney ailment and died of Bright's disease on Sep 14, 1835 at the age of 37.
McDonnell was interested in political economy and wrote 6 books on the subject. He died a wealthy man.
Both McDonell and La Bourdonnais are buried in Kensal Green All Souls' cemetery in London.