Clifford was born in Ballymacelligott, County Kerry, Ireland, on 25 January 1939. He was educated at Clogher National School and St Brendan's College, Killarney. Among his teachers at St. Brendan's was the late Bishop of Kerry, Diarmaid Ó Súilleabháin. From Killarney, he moved to St Patrick's College, Maynooth, where he graduated with a B.Sc. Degree in 1960. After Maynooth, he went to the Irish College in Rome, where he studied for four years and was ordained priest on 22 February 1964. Whilst in Rome, he studied at the Lateran University and obtained a Licentiate in Sacred Theology, being in Rome for the first two sessions of Vatican Council II. As a student, he was given the responsibility of looking after the Irish bishops who stayed in the Irish College.
Clifford's first post after ordination was as a teacher and Dean of Discipline in St Brendan's College, Killarney, where he taught from 1964 to 1972. He commuted to Cork five days per week (1965–1966) for his Higher Diploma in Education. He was later to lecture on a part-time basis in University College Cork in Social Science (1975–1981). He is now a member of the Governing Body representing north and south Tipperary.
Clifford then studied Social Administration at the London School of Economics (1972–1974), where he was conferred with a Master's Degree with distinction. From London, he returned to his native Kerry in August 1974 to become Diocesan Secretary to Bishop Eamon Casey. During that time he also served as Chaplain to St. Mary of the Angels, Beaufort, a home for children with learning disabilities.
The Holy See chose him as Coadjutor Archbishop of Cashel and Emly on 17 December 1985 and he was consecrated on 9 March 1986. The Principal Consecrator was Archbishop Thomas Morris; his Principal Co-Consecrators were Archbishop Gaetano Alibrandi and Bishop Diarmaid O'Súilleabháin, the Bishop of Kerry. He was parish priest of Tipperary town for two and a half years. On 12 September 1988 he was installed as Archbishop of Cashel and Emly in a ceremony in Thurles Cathedral, presided over by the late Cardinal Tomás Ó Fiaich.
A keen footballer in his earlier years, he became the first Kerryman to hold the office of Patron of the Gaelic Athletic Association, in 1989. That same year he was awarded a Ph.D. degree for a thesis on Carers of the Elderly and Handicapped at Loughborough University; this was based on studies he conducted in Kerry just before he left.
Clifford has served on the Emigrant Commission of the Bishops' Conference and he helped to set up the Chaplaincy Scheme to the young emigrants in the USA in 1987.
On 24 March 2010 it was announced by the Holy See that John Magee had formally resigned from his duties as Bishop of Cloyne and was now bishop emeritus and that Clifford, already apostolic administrator there, will remain as such until the appointment of a full-time successor to the Cloyne diocese. Clifford served as Apostolic Administrator to the Diocese of Cloyne until the new bishop, William Crean, was ordained and installed on 27 January 2013.
In October 2010, Archbishop Michael Neary of Tuam, along with Cardinal Seán Brady, Archbishops Diarmuid Martin of Dublin and Dermot Clifford of Cashel and Emly met for high-level talks with heads of Vatican congregations over the apostolic visitation of Irish dioceses in the wake of the Murphy and Ryan reports. While in Rome, the Irish churchmen met with a team of investigators appointed by Pope Benedict to examine the four Irish archdioceses and "some other as yet unspecified dioceses".
The "Apostolic Visitors" included Cardinal Cormac Murphy-O'Connor, Archbishop emeritus of Westminster, inspected Brady's archdiocese of Armagh, and Cardinal Seán O'Malley, Archbishop of Boston, inspected archdiocese of Dublin. Toronto's Archbishop Thomas Christopher Collins investigated Cashel, while Ottawa's Archbishop Terrence Prendergast investigated the western Ireland archdiocese of Tuam. An investigation of the state of Irish seminaries was conducted by Archbishop Timothy Dolan of New York.