Griffith was the eldest son of John Griffith and his wife Elizabeth Bulkeley, daughter of Robert Bulkeley, 2nd Viscount Bulkeley. He matriculated at Christ Church, Oxford on 8 June 1703, aged 16 and was admitted at Inner Temple in 1703. He succeeded to the estate of his father in June 1687. He married Mary Lake, daughter of Sir Bibye Lake, 1st Baronet.
