Leveson was the second son of Sir John Leveson of Halling, Kent, and his second wife Christian Mildmay, daughter of Sir Walter Mildmay. In 1605 he inherited property in Trentham, Staffordshire and Lilleshall, Shropshire from his father’s cousin Sir Richard Leveson, Vice-Admiral of England. Leveson's elder brother, John, who was intended to inherit his father's estates, died in 1612, three years before his father. Leveson therefore inherited his father’s estates in Kent as well as the Vice-Admiral’s estates. He rebuilt the family seat at Trentham Hall between 1630 and 1638 at a cost of over £6000. His Staffordshire relatives included Colonel Thomas Leveson, who held Dudley Castle for Charles I from 1643-46 and was one of 25 former Royalists listed by Parliament in 1651 as subject to 'perpetual banishment and confiscation.' 
In November 1640, Leveson was elected Member of Parliament for Newcastle under Lyme in the Long Parliament. He encouraged Royalist support in Staffordshire and was disabled from sitting in parliament on 24 November 1642 for raising forces against parliament. In 1645 Lilleshall Abbey fell to the parliamentary army and Leveson was imprisoned at the Parliamentary garrison in Nantwich.
Leveson married Katherine Dudley, daughter of Sir Robert Dudley: the couple had no children. Lady Katherine died in 1674, and in her will, endowed almshouses and an educational fund in Temple Balsall. The Lady Katherine Leveson Foundation continues to operate in the 21st century, with the care provider Lady Katherine Housing and Care as the successor to the almshouses, and the Lady Katherine Leveson C of E Primary School taking forward her educational charity. 
Their co-heiresses were the two daughters of his elder brother John. One of them, Christian, married Sir Peter Temple of Stowe. A portrait of this Sir Richard Leveson seems to have found its way into the collection of paintings at Stowe House, Buckingham. A portrait of Sir Richard Leveson, said to be by van Dyck, belonged to the Duke of Sutherland in 1891. This portrait was purchased for £65 02s 00d from the sale of the possessions of the Dukes of Buckingham and Chandos held at Stowe House in 1848. It was described in the sale catalogue as by Van Dyck (whereas other paintings are described as 'after Van Dyck' or a 'copy of Van Dyck'). The catalogue records that Sir Richard is shown ‘in a black dress, with a frill’ and that the painting was bought ‘after a very active competition.’
- The History of Parliament Trust, LEVESON, Richard (1598-1661) retrieved December 2017
- Staffordshire Past Track - Sutherland in Context
- Smith, G (2003). The Cavaliers in Exile 1640-1660 (2014 ed.). Palgrave Macmillan. p. 55. ISBN 1349510718.
- Willis, Browne (1750). Notitia Parliamentaria, Part II: A Series or Lists of the Representatives in the several Parliaments held from the Reformation 1541, to the Restoration 1660 ... London. pp. 229–239.
- "Two charitable sisters". The Foundation of Lady Katherine Leveson.
- Oxford Dictionary of National Biography, 2004, s.n. Sir Peter Temple
- Royal naval exhibition, 1891: the illustrated handbook and souvenir
- Henry Rumsey Forster, The Stowe Catalogue Priced and Annotated 1848, London, David Bogue, Fleet Street, p.178, Item 318