Burgon was brought up in Leeds, where his family have lived and worked for over a hundred years, since arriving as immigrants from Ireland. He was educated at Cardinal Heenan Roman Catholic High School in Leeds. He moved to St Aidan's and St John Fisher Associated Sixth Form in Harrogate to complete his A Levels. He was the first person in his immediate family to go to University, studying English Literature at St John's College, Cambridge. He was chair of Cambridge University Labour Club.
Burgon has said that growing up hearing about the 1984–85 Miners' Strike initiated his interest in politics. His aunt was a member of Women Against Pit Closures and was married to a striking miner. He is the nephew of the former Labour MP Colin Burgon. Tony Benn described meeting Burgon in July 1999 in his diary, describing him as "a good socialist, [who] had written a thesis on 'Tony Benn's influence on the Labour Party'."
Burgon attended the 15 February 2003 anti-war protests against George W. Bush and Tony Blair's planned invasion of Iraq.
In 2004, Burgon stood unsuccessfully as one of the Labour candidates for the Wetherby ward of Leeds City Council.
Burgon qualified as a solicitor in 2006, specialising in employment law for 10 years in the Employment Rights Unit at Thompsons Solicitors in Leeds. Burgon has described how, as a solicitor, he "represented members from a wide range of trade unions in Employment Tribunal cases relating to, for example, unfair dismissal, detriment on trade union grounds, disability discrimination, sex discrimination, discrimination on grounds of religion or belief, unlawful deduction of wages and TUPE."
Burgon was elected as MP for Leeds East at the 2015 general election. He was described by The Observer as one of "the stars of the  intake". Standing on the floor of the House of Commons in May 2015 he prefaced his mandatory oath of allegiance to Elizabeth II by expressing his support for constitutional change for an elected head of state: "As someone that believes that the head of state should be elected I make this oath in order to serve my constituents".
Following the resignation of Ed Miliband as Labour leader, Burgon was one of 10 newly elected Labour MPs who wrote an open letter calling for "a new leader who looks forward and will challenge an agenda of cuts, take on big business and will set out an alternative to austerity – not one which will draw back to the New Labour creed of the past". Along with Jeremy Corbyn, Diane Abbott and John McDonnell, Burgon was one of 48 Labour MPs to defy the whip and vote against the 2015 Welfare Bill explaining that he was voting "for the people of East Leeds and against Conservative attacks on welfare." He was one of 36 Labour MPs to nominate Jeremy Corbyn as a candidate in the Labour leadership election of 2015. Following Corbyn's victory in that campaign he appointed Burgon as Shadow Economic Secretary to the Treasury. He nominated Corbyn again as leader during the 2016 Labour leadership election.
In an October 2015 Channel 4 News interview, Burgon admitted that, despite having been Shadow Economic Secretary to the Treasury for over a month, he was still yet to meet anyone from the city of London's finance and banking industry, nor could he predict the UK budget deficit for 2015.
Burgon is secretary of the GMB Parliamentary Group. In this role, he ensures that issues of interest to GMB members are raised in the House of Commons. Burgon is also a member of the Bakers, Food and Allied Workers' Union. Burgon is the Secretary of the Socialist Campaign Group of Labour MPs, and is regarded as being on the left of the Labour Party. He has taken part in People's Assembly Against Austerity protests. Burgon has called for more democracy within the Labour Party, saying "All parties need to be made more democratic. We've got a membership of well over half a million and I would like the members to have more say in our party's policies and in the way the party's run."
Burgon successfully defended his Leeds East seat in the 2019 General Election (12th December 2019), winning 19,464 votes. This represented 49.8% of the vote, and was -11.5% on the 2017 General Election.
In December 2019 Burgon confirmed he is standing for the deputy leadership of the Labour Party.
Shadow Justice Secretary
Burgon was promoted to Shadow Justice Secretary and Shadow Lord Chancellor on 27 June 2016 following the organised mass resignations in protest against the leadership of Corbyn.
Burgon has called for an end to privatisation in the justice system, labelling privatisation "an ideological experiment ... that has been a calamitous failure", and criticising the part-privatisation of probation by former Justice Secretary Chris Grayling as "a costly failure that has left our communities less safe." As Shadow Justice Secretary Burgon has raised concerns that sustained cuts to the prison service have led to a crisis, with prisons becoming increasingly dangerous for both staff and inmates. He has called for a reversal of Ministry of Justice cuts and for emergency funding to recruit more prison officers and provide prison staff with new equipment, action to end overcrowding and for plans to build new private prisons to be scrapped. At Labour Party Conference in 2017 Burgon announced that a Labour Government would "properly support law centres as engines of empowerment for working-class communities".
In September 2017, the political commentator Iain Dale listed Burgon at Number 86 in 'The 100 Most Influential People on the Left'."
Burgon believes cuts to legal aid are a false economy and deny people justice. Burgon said, "A lack of early legal advice can create unnecessary costs for the taxpayer as legal problems go to court when they could have been resolved earlier or spiral into costly social problems as people lose their homes or jobs ... legal aid cuts have deliberately weakened people's ability to challenge injustices and enforce their rights." In the aftermath of the Windrush Scandal, Burgon criticised Conservative cuts to legal aid for immigration cases in Parliament, saying "The Windrush scandal is one of the cruellest examples of unaccountable state power targeting the vulnerable, the defenceless and the innocent that I can ever remember".
Burgon opposed military intervention and the bombing of Syria following the November 2015 Paris attacks by ISIS militants. Burgon opposed the 2018 Missile Strikes Against Syria. At a protest against the strikes on Parliament Square, he said "The Prime Minister says that bombing of Syria was, in her words, "right and legal". I say it was wrong and immoral. It was wrong and immoral because it risks escalating a war that could involve up to a dozen countries. It was wrong because she never waited for the weapons inspectors to do their job. It was wrong because she never waited for Parliament to have its say."
Burgon is a vocal critic of Donald Trump, commenting during at the Together Against Trump demo during Trump's state visit to the UK that the US President “scapegoats migrants for all society’s ills while letting the powerful off the hook.” He has warned against the consequences of diplomatic tensions between the US and Iran, commenting that Britain could be "dragged into a conflict" and stating that "we don't want to end up being the messengers of Donald Trump".
Burgon has criticised human rights abuses by the Israeli Government in Palestine, and visited the West Bank in March 2016 as part of a group of Labour MPs. Of the visit, Burgon said “I was proud to visit the Occupied Territories of Palestine to show solidarity with the Palestinian people struggling for justice”. On the BBC's Daily Politics show in March 2018 he was asked by Andrew Neil about claims that he had said at a 2016 Labour Party meeting "Zionism is the enemy of peace". Burgon responded "No and that was not my view" and "I didn't make those comments". Footage surfaced in April 2019 of Burgon speaking at an event in 2014 where he was recorded saying "The enemy of the Palestinian people is not the Jewish people. The enemy of the Palestinian people are Zionists, and Zionism is the enemy of peace and the enemy of the Palestinian people", which led to accusations that he lied when he denied that this was a view he held. Burgon said that he had not recalled using what he called a "simplistic" phrase that he would not now use.
Burgon has previously expressed support for the PSUV regime in Venezuela and tweeted a congratulatory message to Nicolás Maduro when he became president in 2013. On an episode of Question Time in February 2019, Burgon refused to express regret for supporting Maduro following the outbreak of the 2019 Venezuelan presidential crisis. He has also expressed support for the Communist Party regime in Cuba and spoke at a memorial event for Cuban President Fidel Castro following his death in 2016. During his speech, in which he praised Castro extensively, he stated that "Fidel Castro was a giant. A man of ideas. A man of action".
Burgon has called climate change "the greatest ever market failure. It is a failure driven by capitalism's endless focus on profits whatever the cost", and has expressed support for the School Strike for Climate. He called Corbyn's speech arguing for Parliament to declare a Climate Emergency "one of the most important ever made in Parliament".
Libel case against The Sun
On 6 February 2019, Burgon won a libel case against The Sun newspaper after it falsely claimed that he had performed with a music group who 'delighted' in Nazi imagery. The High Court ruled that the allegations made by The Sun were defamatory and untrue. Burgon was awarded damages of £30,000, which he said would be spent supporting an apprenticeship in Leeds.