Kemp was educated at Colchester Royal Grammar School. He then served as a soldier and officer in the Royal Anglian Regiment from 1977 to 2006. He started his training as an infantry soldier in 1977 at Bassingbourn Barracks, Cambridgeshire. The following year he entered the Royal Military Academy Sandhurst and was commissioned as second lieutenant on the General List on 5 August 1978. He was posted back to the Royal Anglian Regiment on 3 March 1979, his first regimental appointment was as a platoon commander with the 3rd Battalion, based in Palace Barracks, Belfast. After leaving Belfast he subsequently saw service around the world, including Germany, Kenya and Cyprus. Throughout his military career Kemp completed seven tours on Operation Banner in Northern Ireland, including West Belfast, South Armagh, Derry City and County Londonderry; and a Spearhead tour to Armagh. He was promoted lieutenant on 5 August 1980, and captain on 5 August 1985. He later became an instructor at Bassingbourn before a posting to the 2nd Battalion as the second in charge (2IC) of a Company, Milan Platoon Commander, and then Battle Group Operations Officer, serving across the world including a tour as part of UNFICYP in Cyprus.

Kemp spent much of his early career working in mechanized and armoured infantry, including command of a Warrior company and command of the Armoured Infantry Training and Advisory Team covering British military units in Germany and the UK. He wrote the military operations manual, 'Armoured Infantry Company Group Tactics', which remains in use today forming a part of British Military doctrine.As a captain in the Royal Anglian Regiment he took part in the first Gulf War, in 1990–91, commanding Brigadier Patrick Cordingley's 7th Armoured Brigade Tactical Headquarters. He was promoted major on 30 September 1991. In the late 1990s, he devised a new system for training Warrior AFV crews in gunnery combined with tactical movement, utilising standard armour ranges. The system was a development of US armoured infantry training techniques, and was subsequently adopted by elements of the Royal Armoured Corps.

Following tours with the headquarters of the 7th Armoured Brigade, at Camberley Staff College and in Headquarters Northern Ireland, he returned to the 2nd Battalion of the Royal Anglian Regiment as Point Company Commander, leading the company in Bosnia, Germany and Canada. In Bosnia, he served on operations with the United Nations (UNPROFOR). He then took over the Armoured Infantry Training and Advisory Team based at Hohne and Sennelager, and held a staff appointment in the Ministry of Defence. He was promoted lieutenant-colonel on 30 June 1997, and commanded the 1st Battalion, Royal Anglian Regiment in Oakington and Londonderry from 1998 to 2000. After this command, he was appointed Commander of the Operational Training and Advisory Group and then spent six months as Counter Terrorism and Security Advisor to the Government of Macedonia.

Cabinet Office

Kemp was attached to the Cabinet Office from 2001–2006, with a six month posting to Afghanistan in 2003. At the Cabinet Office he worked for the Joint Intelligence Committee and the national crisis management group, COBR, during the period of the 2005 London bombings and a series of high profile kidnappings of British nationals in Iraq and Afghanistan. While working at the Cabinet Office his responsibilities included Iraq, and he made several visits to Baghdad, Fallujah and Mosul. He was involved in devising and developing government strategies and policies on counter terrorism. He was promoted colonel on 30 June 2004, and retired from the army on 30 March 2006.

Afghanistan

Kemp was Commander of British Forces in Afghanistan in 2003. He was responsible for all British forces deployed in Afghanistan, including a patrols unit operating under ISAF in Kabul, a training group working within the US-led Afghan National Army training mission, and a Provincial Reconstruction Team based at Mazar e-Sharif. During his tenure he also acted as Military Attache, working closely with the British Ambassador to Afghanistan and with the US Task Force Commander at Bagram and the UN Secretary General's Special Representative.

Kemp set up and led an ad hoc joint counter-terrorist operations group comprising forces from the US Marine Corps 4th Marine Expeditionary Brigade (Antiterrorism) and the British Army, which achieved a number of successes against Al Qaeda-associated extremists in Kabul. While Commander in Afghanistan, he devised a doctrine for combating suicide attack, which did not previously exist in the British forces. That doctrine was subsequently taken into general use in the armed forces, promulgated by the Permanent Joint Headquarters.

Richard Kemp led a campaign in 2007 and 2008 to recognise the sacrifice of British troops killed and wounded in action by the award of a medal similar to the US Purple Heart. The campaign gained the support of the majority of Members of Parliament, through an Early Day Motion in the 2008–2009 parliamentary session; as well as widespread support among serving and retired military ranging from Private Harry Patch, last survivor of the Battle of Passchendaele to Field Marshal Sir Edwin Bramall, former Chief of Defence Staff.

The campaign also prompted specific debates in the House of Commons and the House of Lords, and led to the MOD announcement in 2008 of a new award to be made to the next of kin of British soldiers killed in action, which was launched in July 2009. The award, called the Elizabeth Cross, is the first decoration to be named after a reigning monarch since the institution of the George Cross in 1941. The award was back-dated to the end of the Second World War.

The Government has so far rejected demands for a medal to acknowledge the sacrifice made by service men and women seriously wounded in action.

Kemp has disputed allegations that Israel engaged in war crimes and crimes against humanity during Operation Cast Lead in December 2008 to January 2009

In June 2009, Kemp wrote a document entitled "HAMAS, THE GAZA WAR AND ACCOUNTABILITY UNDER INTERNATIONAL LAW" for the Jerusalem Center for Public Affairs, an NGO focused on Israeli security needs headed by former Israeli ambassador Dore Gold. Kemp, in his article, claims that Hamas discarded their uniform once fighting began and deployed women and children as suicide attackers. Kemp also claims that during the Gaza War that "women and children are trained and equipped to fight, collect intelligence and ferry arms and ammunition between battles".

In October 2009, Kemp made a presentation to the United Nations Human Rights Council in response to the introduction of the Goldstone report that accuses Israel and Hamas of committing war crimes and possible crimes against humanity during the Gaza War. Kemp spoke to the UN HRC on behalf of UN Watch. While Kemp acknowledges that "of course innocent civilians were killed. War is chaos and full of mistakes. There have been mistakes by the British, American and other forces in Afghanistan and in Iraq, many of which can be put down to human error. But mistakes are not war crimes.", Kemp states the following: "based on my knowledge and experience, I can say this: during operation Cast Lead, the Israeli Defense Forces did more to safeguard the rights of civilians in the combat zones than any other army in the history of warfare".

Kemp has also claimed to have relied on Israeli expertise in Afghanistan, stating that after finding himself unable to deal with the threat of suicide bombers, he received help from an Israel Defense Forces (IDF) anti-suicide bombing expert. He also claimed to have received help from the Mossad, Israel's intelligence agency, following the 2005 London bombings, and is well-acquainted with Israeli intelligence from past cooperation. Kemp has been highly critical of what he calls "the automatic assumption that everything the IDF does is illegal", stating that "there is harsh criticism against the US and British armies as well, but they are given the benefit of the doubt. Israel is always automatically condemned, no matter what. It's a joke. Even the conspiracy theory that the one responsible for the September 11 attacks is not al-Qaeda but Israel refuses to die out, it's unbelievable".

Kemp co-wrote a book called Attack State Red about the deployment of the Royal Anglian Regiment in Helmand Province, Afghanistan, in 2007 as a part of Operation Herrick. The title of the book comes from the British military standing operating procedures (SOPs) for attacking compounds and buildings in Afghanistan. Kemp wrote the book with Chris Hughes, a journalist with the Daily Mirror. The book seeks to place the civilian reader into the boots of the fighting soldier. It has been written with the intention of providing an insight into what it is like to confront an enemy in harsh battle conditions. The book was published in September 2009 by Penguin Books.

Kemp was appointed Member of the Order of the British Empire (MBE), Military Division, on 25 April 1994 in recognition of his intelligence work in Northern Ireland in 1993, and was awarded the Queen's Commendation for Bravery for service as a commander in the United Nations Protection Force in Bosnia in 1994. He was promoted Commander of the Order of the British Empire (CBE), Military Division, in the New Year Honours 2006.