A native of Fairview Park, Ohio, Bishop Griffin was ordained a priest by John Cardinal Krol on May 28, 1960, in St. John Cathedral, Cleveland, Ohio.

Bishop 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 in Canon Law (J.C.L., magna cum laude) from the Pontifical Lateran University in Rome in 1963. Upon his return to the Cleveland Diocese, Bishop 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 (J.D., summa cum laude) in 1972. He passed the Ohio Bar and is a licensed attorney. Bishop 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, Ohio. In June of 1979, he was appointed Auxiliary Bishop of the Diocese of Cleveland. Pope John Paul II appointed Bishop Griffin the tenth Bishop of the Diocese of Columbus, Ohio on February 7, 1983. Bishop Griffin was installed as Bishop of the Diocese of Columbus on April 25, 1983.

While serving as Bishop of Columbus, James A. 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).