Beesley was born on 16 April 1971 in Burnage, Manchester. His father, Maxton Beesley Sr, is a professional jazz drummer and actor, and his mother was a jazz singer who performed under the stage name Chris Marlowe. He has a step-brother, Jason Milligan, who is also an actor. He has a brother Gary who is a teacher. His parents divorced when he was a baby. His middle name was inspired by the American actor Gig Young. His father was also a regular guest on the Ted Rogers show 3-2-1, doing impressions amidst several sketches.

Beesley was a member of the jazz band Incognito, but in 1994, after an eight-month tour with them, decided to end his music career and start acting. On 28 October 2006, Beesley appeared as a special guest pianist for James Brown during his performance at the Roundhouse in London as part of the BBC’s Electric Proms festival. In 2020 he recorded his first solo album titled ‘Groove Spectrum’ at Capitol studios in Los Angeles with legendary Grammy winner, producer, mixer, Al Schmitt mixing the Album. It features Paul Weller, Robbie Williams, Fred Durst, Lisa Stansfield, jazz pianist Christian Sands, Tommy Blaize, Antonique Smith, John Turrell, Omar, and Francci Richards. A multi instrumentalist Beesley wrote, played, and scored the album.

Beesley has performed in concert as a percussionist and pianist for Robbie Williams, Take That, Jamiroquai, and the Paul Weller Movement. He played percussion and piano at Williams’ Knebworth gigs in summer 2003 and at Hyde Park for Live 8 London. Williams’ first concert of his Close Encounters Tour in Perth on 30 November 2006 saw Beesley as percussionist for the second half of the show. He also played drums during the performance of “Rudebox” at Williams’ second Brisbane concert and during a BBC Electric Proms show.

In June 2007, Beesley starred in the ITV series Talk to Me. He also starred in the film Red Roses and Petrol alongside Malcom McDowell, and in the BBC thriller The Last Enemy with Benedict Cumberbatch and Robert Carlyle. In October 2008, he was one of the lead characters in BBC One’s remake of the 1970s drama Survivors, as remorseless former criminal Tom Price. The series consisted of six episodes, with a second series of six episodes transmitted in January 2010. He starred in the ITV drama The Reckoning in April 2011. Beesley gained more exposure with American audiences with a recurring role on Suits in 2013 and from October 2015 with another recurring role, this time in the US spy drama Homeland opposite Claire Danes, as ex-British special forces bodyguard Mike Brown. From 2016 to 2018 he starred in Jamestown as Henry Sharrow, depicting the first British settlers to colonise Virginia in 1609. He joined the cast of The Outsider on HBO in June 2019, which aired from January 2020.

Beesley has been the narrator for several TV advertisements, including ITV, NFU Mutual’s insurance, Energizer’s lithium batteries, and PC World. He also provided the voiceover for Manchester United’s season review DVD. In 2011, he appeared in a TV campaign for the UK recruitment company Jobsite, and was also featured on their homepage. Beesley narrated the documentary Chatsworth, shown on BBC One in May 2012.