He was the first and only Vicar Apostolic of the London District educated wholly in England. At the age of thirteen he was sent to St. Edmund's College, Old Hall, where he went through the whole course, and was ordained priest in 1814. Four years later he was chosen as president, at the age of 27. He ruled the college for fifteen years, at the end of which time he was appointed coadjutor to Bishop Bramston, then Vicar Apostolic of the London District. He was consecrated as titular Bishop of Olena at St. Edmund's College, 28 October, 1833. Within three years Bishop Bramston died, and Bishop Griffiths succeeded him.
The agitation for a regular Catholic hierarchy in England became more and more pronounced and as a preliminary measure, in 1840, the four ecclesiastical "districts" into which England had been divided since the reign of James II of England were subdivided to form eight, Dr. Griffiths retaining the new London District. Soon after this, the Oxford movement and attendant Catholic conversions began: and the immigration of Irish Catholics grew. At the same time the growth of the British colonies, many of which had been until lately ruled as part of the London District, brought him into contact with the government. In all these spheres Griffiths discharged his duties with practical ability; but it was thought that he would not have the breadth of view or experience necessary for initiating the new hierarchy, and (according to Bishop Ullathorne) this was the reason why its establishment was postponed.
When Griffiths died, somewhat unexpectedly, in 1847 Ullathorne himself preached the funeral sermon. The body of the deceased prelate was laid temporarily in the vaults of Moorfields Church; but two years later it was removed to St. Edmund's College, where a new chapel by Augustus Pugin was in course of erection, and a special chantry was built to receive the body of Dr. Griffiths, to whose initiative the chapel was due. An oil painting of Dr. Griffiths is at Archbishop's House, Westminster; another, more modern, at St. Edmund's College.