The fifth of seven children, Griffin was born in Fairview Park, Ohio, to Thomas Anthony Griffin and Margaret Mary Hanousek. He attended St. Angela Merici School  in Fairview Park, and St. Ignatius High School in Cleveland. He attended St. Charles College in Catonsville, Maryland, and Borromeo College in Wickliffe, Ohio, where he received his Bachelor of Arts degree in Philosophy. He attended St. Mary Seminary in Cleveland. Bishop John Krol ordained him to the priesthood on May 28, 1960, in St. John Cathedral.
Griffin spent one year as Associate Pastor at St. Jerome Parish in Cleveland. In 1961, he was sent to Rome to pursue graduate studies in Church Law. He received his Licentiate of Canon Law magna cum laude from the Pontifical Lateran University in 1963. Upon his return to the Cleveland Diocese, Griffin served as Secretary-Notary of the Marriage Court of the Cleveland Diocese for two years. He was appointed to the Chancery Staff in 1965, and served as Associate Chancellor and Vice Chancellor. During that time, he attended night classes at Cleveland State University and received his Doctorate in Civil Law summa cum laude in 1972. He passed the Ohio Bar and is a licensed attorney. Griffin served as Chancellor of the Diocese of Cleveland from 1973 until 1978.
In January 1978, he was appointed by Bishop James Hickey as Vicar General of the Cleveland Diocese and Administrator pro tem of St. John Cathedral in Cleveland. In April of that year, he was appointed pastor of St. William Parish in Euclid. In June 1979, he was appointed Auxiliary Bishop of the Diocese of Cleveland. Pope John Paul II appointed Griffin the tenth Bishop of the Diocese of Columbus, Ohio on February 7, 1983. He was installed as Bishop of the Diocese of Columbus on April 25, 1983.
While serving as Bishop of Columbus, Griffin established the Foundation of the Catholic Diocese of Columbus (1985); inaugurated far-reaching planning processes for the Diocese; initiated the Legacy of Catholic Learning (1989) and Challenge In Changing Times (2000) campaigns to help meet the educational and future needs of the Diocese; and helped initiate the community-wide, faith-based task force titled "Breaking The Silence" that continues to work to reduce family violence. Bishop Griffin also served on a number of committees of the United States Conference of Catholic Bishops and was President of Catholic Relief Services (1991–1995).
Bishop Griffin announced his retirement on October 14, 2004, citing limitations brought on by age and arthritis that detracted from the energy necessary to serve the Church of Columbus as he had been accustomed. He announced concurrently that John Paul II had selected Bishop Frederick F. Campbell of the Archdiocese of St. Paul and Minneapolis as his successor.