Campbell was born on a farm at Rathfriland, near Newry, County Down, Northern Ireland, and was the youngest of four boys. Campbell was educated at St Colman's College, Newry. After St Colman's he attended St Joseph's Seminary in Belfast, part of the philosophy faculty at Queen's University Belfast and attended postgraduate studies at the KU Leuven, where he considered the priesthood. He attended the University of Pennsylvania on a Thouron Award Fellowship.
He was a lector at Westminster Cathedral, and, at one time, was an Extraordinary Minister of Holy Communion. In 2010, “The Tablet” named him as one of Britain’s most influential Roman Catholics.
Campbell joined the British Diplomatic Service in 1997, aged 27. His early career focused on Europe with postings to the European Enlargement Unit of the Foreign and Commonwealth Office and secondment to the European Commission, as part of its Delegation to the United Nations in New York.
From 1999–2003 he worked at 10 Downing Street, appointed first as Policy Adviser to the Prime Minister (1999–2001) and then Private Secretary to the Prime Minister (2001–2003). From 2003–2005, he was First Secretary at the British Embassy in Rome and, subsequently, on sabbatical, Senior Policy Director with Amnesty International in London. He was Ambassador to the Holy See from 2005 to 2011 and was then appointed Deputy High Commissioner to Pakistan, based in Karachi. In 2013 Campbell was appointed Head of the Policy unit for the FCO and also Head of Innovation for UKTI. In 2014 he became Vice-Chancellor of St Mary's University, Twickenham, London.
In August 2014 in light of attacks in Iraq, Campbell, criticised the Government for its failure to speak out over what he described as ethnic cleansing. “A culture and civilisation is being destroyed and our political leaders are silent,” he said, asking Prime Minister David Cameron via the social networking site Twitter: “Why is the UK silent on the ethnic cleansing of Christians from Mosul?” A Foreign Office spokesman told The Tablet: “The threat to Christians in the Mosul area is a particular tragedy, given that it has one of the oldest Christian communities in the world. We will work with the new Baghdad Government to raise these matters further.”
Airing in February and March 2010, a 3-part documentary by BBC Northern Ireland called Our Man In The Vatican, focused on a year in the life of Ambassador Campbell. Contributors to the programme included Tony Blair, the Prime Minister during Campbell's role as Private Secretary.
Campbell has stated that he is both "British and Irish" and that he is not "going to exaggerate one identity at the expense of the other."