He was educated at St Paul's School, but ran away from home and made his first appearance on the stage as Tressel in Richard III at the Old Orchard Street Theatre in Bath in 1791.

In the same year he married Elizabeth, the sister of Mary Ann Rundall, and they would in time have ten children.

At the Surrey, where he was the lessee first from 1806–14 and then again beginning in 1827, to avoid the patent restrictions on drama outside the West End, he presented Shakespeare and other plays accompanied by ballet music.

Leigh Hunt compared him favorably as an actor with David Garrick; Lord Byron thought him inimitable in high comedy; and Macready praised his versatility.

Elliston was the author of The Venetian Outlaw (1805), and, with Francis Godolphin Waldron, of No Prelude (1803), in both of which plays he appeared.