Paxman was educated at Malvern College and New College, Oxford, and joined the Civil Service in 1974. He worked in the Department of the Environment from 1974 to 1976.
From 1978 to 1979, he attended the École nationale d'administration, Paris. Afterwards, he worked for the Department of Transport, until 1980.
From 1980 to 1984, Paxman served as the United Kingdom's Permanent Representative to the EC for Environment, Consumer Affairs and the Internal Market in Brussels.
During the mid-eighties, Paxman worked for the Foreign and Commonwealth Office, first in the European Community Department (External) between 1984–1986, then in the Southern European Department, between 1986-1988.
In 1988, he was appointed Head of Chancery at the British High Commission in Singapore, a position he held until 1991, when he took a position as Deputy Head of the Assessments Staff in the Cabinet Office, until 1993.
Between 1994 and 1998, Paxman was the Counsellor of Economic and Commercial Affairs in Rome.
Between 1999 and 2002, Paxman worked with the UK Permanent Representation to the EU in Brussels, after which, in 2002, he was appointed Minister and Deputy Head of Mission at the UK Embassy in Paris and Counsellor of Political and Institutional Affairs.
On 16 June 2005, the UK Government announced that Paxman had been appointed Ambassador to the United Mexican States (Mexico). He assumed the role as the Ambassador on 3 October 2005. Only weeks later, on 25 October, he travelled to Cancún to oversee the evacuation of British nationals affected by Hurricane Wilma.
Paxman became Ambassador to Spain on 19 October 2009. He was appointed Companion of the Order of St Michael and St George (CMG) in the 2013 New Year Honours "for services to UK interests in Spain and Mexico." He was replaced by Simon Manley in October 2013.
Paxman is married to Ségolène Cayol. He is a Lieutenant of the Royal Victorian Order. With his wife, he has three daughters, Julia, Lauren and Alice, born in 1982, 1984 and 1988. His older brother Jeremy is a journalist for the BBC.