He was the son and heir of Sir John Conway of Arrow, and his wife Ellen or Eleanor, daughter of Sir Fulke Greville of Beauchamp's Court, Warwickshire and his wife Elizabeth Willoughby, 3rd Baroness Willoughby de Broke.
[2] He then served as governor of Brill, an English Cautionary Town near Rotterdam in the Netherlands, where his daughter Brilliana (who married Robert Harley) was born.
Conway was appointed to the Privy Council in 1622 and made a Secretary of State in January 1623 for five years.
No doubt as a result of his time in the Netherlands, he was a supporter of a 'Protestant' foreign policy; he was sent as ambassador to Prague.
Conway married firstly Dorothy (died 1613),[3] daughter of Sir John Tracy of Tedington, Gloucestershire, and widow of Edmund Bray.