Griffith is educated at Trinity College Dublin, and gains a licence in civil engineering in 1868. He serves a two-year apprenticeship under Dr. Bindon Blood Stoney, the Engineer in Chief of the Dublin Port and Docks, before working as assistant to the county surveyor of County Antrim. He returns to Dublin in 1871 and works as Dr. Stoney’s assistant, becoming the Chief Engineer in 1898 before retiring in 1913.
Griffith serves as president of the Institution of Civil Engineers of Ireland between 1887 and 1889 and of the Institution of Civil Engineers between 1919 and 1920. He is elected Commissioner of Irish Lights in 1913 and is a member of the Royal Commission on Canals and Waterways between 1906 and 1911.
Griffith purchases and drains the bogland at Pollagh, part of the Bog of Allen. A peat fueled power station is built which drives an excavator and excess peat is taken via the Grand Canal for sale in Dublin. The site is sold to the Turf Development Board in 1936 who use it as a basis for all of their later peat fueled power stations. The area is now a nature reserve.
Griffith receives a knighthood in 1911 and becomes vice-president of the Royal Dublin Society in 1922. He serves as Honorary Professor of Harbour Engineering at Trinity College, his alma mater, and receives an honorary M.A.I. degree from the University of Dublin in 1914. From 1922 he is an elected member of the Seanad Éireann, the Irish Free State senate, until its abolition in 1936. In the 1930s he and Sarah Purser endow the Purser Griffith Travelling Scholarship and the Purser Griffith Prize to the two best performing students in European Art History at University College Dublin.
Griffith dies at Rathmines Castle in Dublin on October 21, 1938.