Son of Alexander Erskine (d. 1756), 5th Earl of Kellie, and Janet Pitcairn (d. 1775). Brother of Elizabeth, Anne, Jenny, Archibald and Andrew Erskine.
Erskine was a composer of some fame in his own time, as well as being an able violinist who directed the concerts in Saint Cecilia's Hall in Niddry's Wynd, Edinburgh in the 1760s. Grand Master of the Masonic Grand Lodge of Scotland (1763-65).
He died in Brussels on October 9th, 1781 and was succeeded as Earl of Kellie by his brother Archibald.
Boswell knew Thomas Alexander through his brother Andrew Erskine. On October 20, 1762, Boswell borrowed five guineas from him. On May 26, 1763, Boswell took him on a visit to Lord Eglinton's in London.
Most of Erskine's musical works seems to have been lost, but some 1839 editions of Minuets & Songs, Composed by the Right Honourable Thomas Earl of Kelly are sometimes available from úsed bookstores such as AbeBooks, as is his Trio Sonata in F for 2 Violins & Continuo in a 1991 PB edition.