Aston Cockayne was the son of Thomas Cockayne and Ann, the daughter of Sir John Stanhope; Cockayne was born at Ashbourne Hall in Derbyshire, and baptised on 20 December 1608. He was educated at Trinity College, Cambridge, the University of Oxford, and at the Inns of Court. Like many other aristocrats of his time, he travelled through Europe in his youth, spending much of 1632 in France and Italy; like a few, he became fluent in their languages, and translated works of literature into English.