Doran was born in Dublin in 1953. He was educated at Christian Brothers College, Monkstown Park where his father also taught. Doran began his seminary formation from 1970 to 1974 in Holy Cross Seminary, Dublin and from 1974 to 1977 in the Pontifical Irish College, Rome. He was ordained a Deacon on 6 May 1976 in the Pontifical Irish College by the late Bishop Dominic Conway of Elphin. Doran attended University College Dublin, from 1971 to 1974 was awarded a Bachelor of Arts in Philosophy and French. He attended Pontifical Gregorian University from 1974 to 1977 where he earned an STB Theology having already studied philosophy. He was ordained a priest on 6 July 1977 for the Archdiocese of Dublin. From 1978 to 1979 he studied at UCD where he earned a Higher Diploma in Education.

After ordination, until 1983 he was a Teacher in the vocational school in Ringsend, Dublin. From 1980 to 1983 he served in the diocesan Secretariat for Education. In 1983 he was appointed University Chaplain. He furthered his studies at University College Dublin, from 1985 to 1987 where he earned an MA in Philosophy.

In 1991 he returned to Rome, earning a doctorate in philosophy at the Pontifical University of Saint Thomas Aquinas, Angelicum which he completed in 1995, at the same time carrying out the Service Spiritual Director at the Irish College.

He returned to Ireland and was Parochial Vicar at Foxrock Church from 1995 to 1998, and Director of the diocesan, and then national programme for vocations from 1998 to 2006. He was appointed parish priest of Glendalough in 2005, serving until 2009. He was appointed Secretary General of the Preparatory Committee of the 50th International Eucharistic Congress (2008–2012).

In 2005 Doran, in his capacity as a board member of the Mater Hospital, was incorrectly reported to have been a member of a three-person subcommittee, along with a nun, Sister Eugene Nolan, who delayed the trial of a new lung cancer drug because female patients taking the drug would have been required to take birth control to avoid birth defects, contrary to the hospital's catholic ethos.

In October 2013 he resigned from the board of the Mater Hospital after management confirmed it would comply with the new Protection of Life During Pregnancy Act.

He was ordained as bishop on 14 July 2014 by Bishop Christopher Jones with the Principal Co-Consecrators being Archbishop Michael Neary of Tuam and Archbishop Charles John Brown. In February 2017, the Irish Catholic newspaper announced that Bishop Doran had reached an agreement for a new community of religious sisters to establish a house in his diocese. These are members of the Servant Sisters of the Home of the Mother congregation, and would settle at Sacred Heart Parish, Roscommon.

In an interview with NewsTalk Breakfast radio in March 2015, Doran differentiated between Marriage and same-sex relationship: "One […] is of its very nature, directed towards the upbringing, the care of children, and one which isn't and they can't be said to be the same." and that some people who have children "are not necessarily parents". The Archbishop of Dublin, Diarmuid Martin, subsequently described it as "an unfortunate phrase" and hoped that people were not offended by Doran's comments. Doran later said that he regretted the hurt experienced by people either due to his choice of words or to the manner in which what he said had been subsequently reported.