Murray was born in Edinburgh, to a cooper father and shop assistant mother in 1976. Brought up in the Wester Hailes area of Edinburgh, he attended Dumbryden Primary School, then Wester Hailes High School.
Upon completing his secondary school education, Murray read Social Policy and Law at the University of Edinburgh's Academy of Government. He graduated with an honours degree (BA) aged twenty. While studying at university, he had a part-time job in a local fish & chip shop before setting up and running a pizza delivery service.
After graduation, Murray worked for Royal Blind in pensions management, before being head-hunted by an Edinburgh-based internet television station (Worldart.com) during the dot-com boom where he helped to build a new online TV station. Despite his efforts, the company ran out of funding and he was made redundant; he then founded his own event management business (100 mph Events Ltd). Murray also organised a student exchange programme in Nepal to fund school buildings and staff.
In 2003, Murray stood in the council elections for Liberton winning the seat for Labour at the age of 27; he later represented the larger Liberton and Gilmerton ward from 2007 to 2010.
Murray was returned for the seat of Edinburgh South at the 2010 general election. He served on the Business, Innovation and Skills Select Committee and the Environmental Audit Select Committee. In 2011, he was appointed to the Official Opposition frontbench.
At the 2014 Scottish independence referendum, Murray campaigned against independence. He claimed to have encountered hostility from independence activists and reported that his office premises had been plastered with pro-independence "Yes" stickers, which were immediately removed.
Murray was re-elected as MP for Edinburgh South at the 2015 general election, with an increased share of the vote and an increased majority. He was the only Labour MP returned for a Scottish constituency.
He was appointed as Shadow Secretary of State for Scotland on 11 May 2015 by Acting Labour Party Leader Harriet Harman. He was re-appointed to the same role by newly-elected Labour Party Leader Jeremy Corbyn in September 2015.
Days after the EU referendum, on 26 June 2016, he resigned around the same time as some other members of the Shadow Cabinet, citing a lack of confidence in Corbyn's leadership ability to win a general election. He then nominated Owen Smith in his failed leadership challenge against Corbyn. After Corbyn's successful re-election as Labour leader with an increased majority, Murray said he would only return to Corbyn's frontbench if he reinstated Shadow Cabinet elections and stopped using the threat of deselection to enforce loyalty.
Murray has accused Corbyn of being "all over the place" on potential Labour Party co-operation with the SNP. His replacement as Shadow Scottish Secretary Dave Anderson refused to rule out a deal with the SNP at Westminster.
Prior to the 2019 general election, Murray faced the threat of a reselection contest forced by Unite the Union. Local members refused to back such a contest so it could not proceed. Following the election, he again became Labour's only MP in Scotland, as the party's vote share decreased by 8.5%.
On 7 January 2020, Murray announced that he would run for Deputy leader of the Labour Party in the deputy leadership election.
Murray is a member of Progress.