Coburn was born in Glasgow, Scotland.
Coburn is openly gay and speaks fluent French and conversational Arabic He had worked as an art dealer and City of London trader before owning a freight company.
Coburn contested the South-east London constituency of Old Bexley and Sidcup in 2010, and finished in fifth place with 1,532 votes. He also stood in Bexley and Bromley at the 2012 London Assembly election, finishing in fourth place.
Coburn ran in the 2014 European Parliament election to become a Member of the European Parliament for Scotland, as a member of UKIP Scotland. At the end of 2013, UKIP Scotland was dissolved after infighting tore the regional party apart; the party's administrative body was dissolved, Mike Scott-Hayward (the chairman and chief fundraiser) quit, and UKIP leader, Nigel Farage, fired Lord Christopher Monckton via email. The main party as a whole, and UKIP Scotland focused on supporting the candidates for the upcoming European elections. After Coburn won the seat, he was appointed leader of UKIP Scotland. His victory made him UKIP's second openly gay MEP, after Nikki Sinclaire.
In July 2016, Coburn declared his support for Steven Woolfe in UKIP's leadership election to replace Nigel Farage.
At the 2017 snap general election, Coburn unsuccessfully ran for the Kirkcaldy and Cowdenbeath constituency, securing just 1.2% of the vote and losing his deposit in the race.
In the face of local concerns Coburn has come out in favour of fracking in Falkirk, the constituency he stood for in the 2015 General Election.
In 2015, Coburn compared Scottish government minister Humza Yousaf to convicted terrorist Abu Hamza, later apologising and calling it a "joke". The then UKIP leader Nigel Farage called it a "joke in bad taste" and EU President Martin Schulz said that whereas the remarks were "not worthy of any elected member", he could not act on remarks made outside parliament. Parliamentary candidate Tim Wilson quit the party in response, saying "he had been "systematically gagged by the party whip and forbidden to speak about Islam favourably".
In April 2015, a Wikipedia account verified as operated by Coburn's office was blocked indefinitely for edit warring over Coburn's Wikipedia article. Coburn claimed he had directed one of his staff to make the changes in order to clear the page of “garbage” and “nonsense”.