Assheton was the son of Sir Ralph Assheton, 1st Baronet of Lever, Lancashire and his wife Dorothy Bellingham, daughter of Sir James Bellingham of Levens, Westmorland.
He succeeded his father to the Assheton baronetcy on 18 October 1644 and to the family's Downham estate near Clitheroe.
(Great Lever, the family's original home, had been sold by his father in 1629.
[1] He built the vault beneath the Assheton chapel at St Leonard's Church, Downham and endowed a sermon on the resurrection to be preached an annually on the anniversary of his death.
[3] He married twice; firstly Lady Dorothy Tufton, the daughter of Nicholas Tufton, 1st Earl of Thanet and secondly Elizabeth, the daughter of Sir Sapcotes Harrington of Rand, Lincolnshire with whom he had one son, who died before him.