He was the son of Richard Burke, 4th Earl of Clanricarde, by his wife Frances Walsingham. In 1622, Ulick married his only wife Anne Compton, the daughter of William Compton, 1st Earl of Northampton and his wife Elizabeth Spencer. They had a single child, Margaret Burgh who married Charles MacCarty, Viscount Muskerry. In 1652, he made peace with the victorious Oliver Cromwell. He lost his lands in the Act of Settlement 1652 but his heirs regained them after the Restoration of Charles II in the Act of Settlement 1662.