So the claim is that your James William Domville (1817 - abt. 1883) is a descendant of Peter Domville and his wife Margaret Carrington (who have profiles on WikiTree). As your article states, a descent can be found from Margaret Carrington through Warburton-Stanley-Goushill-FitzAlan to Henry III, king of England.
However, you may have a problem connecting James William Domville to Peter Domville. The dumville.org site says that:
“Edmund Dumbile was a hatmaker in Olney, a town in north Buckinghamshire, in the middle 1700s. Nothing is known of his parentage or birthplace. The earliest record available on Edmund comes from the Olney Parish Registers (1665-1812). An entry on November 30, 1731 has the christening of "John Dumbile, the son of Edmund", and later, a christening of "Elianor, daughter of Edmund and Sarah Dumbile" on April 12, 1734. There is no further record of Elianor and some confusion over John, as mentioned in a subsequent paragraph.”