Skinner grew up in London and Tunbridge Wells, and attended Eastbourne College from 1998 to 2003. He graduated from the London Academy of Music and Dramatic Art (LAMDA) in 2006.
Skinner has worked extensively in film, television, and theatre since 2007. His first professional theatre role was in the English Touring Theatre's 2007 production of French Without Tears. In addition to performing in many different theatres in London and around the United Kingdom, he played supporting roles in two BBC series: Felix, the brother of Angel Clare, in the 2008 adaptation of Tess of the D'Urbervilles, and Lionel, the estranged son of the main character Logan Mountstuart, in the 2010 production of Any Human Heart. He also played the role of Joly, one of the student Revolutionaries, in the 2012 film of Les Miserables.
On 7 October 2013, it was announced that Skinner would play the role of Luis Carruthers, a closeted gay man who is in love with in the show's protagonist, Patrick Bateman, in the world premiere of American Psycho at the Almeida Theatre. He also appeared on the London cast album, which was released in 2016.
Skinner continued to work in both television and on stage in 2014, playing the role of Dr. Barnaby Ford in the BBC series Our Zoo, and appearing at the Theatre Royal, Bath as Camille in Helen Edmundson's adaptation of Therese Raquin, and Yephidikhov in Simon Stephens' new translation of The Cherry Orchard at the Young Vic. He returned to the Young Vic in the summer of 2015 to play dual roles in Nick Gill's adaptation of The Trial.
On 10 September 2015, it was announced that he had been cast as Unwin Trevaunance, an aspiring Member of Parliament, in the second series of the BBC production of Poldark, which aired in 2016. That year, Skinner was also featured in two different comedy series: The Windsors, a spoof of the royal family that aired on Channel 4, where he played Prince william, and Fleabag, a BBC3 and Amazon production, where he played the protagonist's hapless boyfriend Harry.
In 2017, Skinner also appeared in Star Wars: The Last Jedi.
