Alexander was born in Sheffield, Yorkshire. His mother, Vicki Thornton, was one of the founders of the Coleford Music Festival. Alexander attended St John's Primary School in Coleford and Monmouth Comprehensive School leaving to study Performing Arts at Hereford College of Arts. While at Monmouth Comprehensive School, he acted in two school plays: Guys and Dolls, as Benny, and The Caucasian Chalk Circle, as the Corporal. The New York Times reported that he wrote his first song on his father's Casio keyboard aged 10. His parents separated when he was 13 and he was brought up by his mother as a single parent.
He obtained the services of an agent when he was 16 while auditioning for a part in UK TV series Skins. In a YouTube video uploaded from NylonMagazineTV, Alexander said he dropped out of school. This can be clarified that he actually dropped out of Hereford College of Arts to pursue his acting career as parts were offered: "I started acting when I was young; it just sort of happened. I dropped out of school to work around the world, which was amazing." "I really wanted to become a singer or a musician ... All of a sudden I was an actor. It was never something I’d set my heart on being. I’m still trying to work it out ... I really hated school because I was totally bullied. But you’re never bullied in drama class because the weird kids do well in drama class. That’s a safe place". Alexander was interviewed about his mental health by Guardian columnist Owen Jones.
Alexander's acting career began in the film Summerhill released in 2008. His next film Bright Star was nominated for an Academy Award in the United States (for Best Achievement in Costume Design). He acted in the 2009 released films Tormented starring Alex Pettyfer and Enter the Void. In 2010 he played Evan in the Bush Theatre production of The Aliens. Alexander contributed to the script and music for indie film The Dish & the Spoon, released in early 2011. In 2012, he appeared in the theatre production of Mercury Fur, taking the part of Naz at The Old Red Lion, Islington.
From March to June 2013, Alexander starred as Peter Pan in the West End play Peter and Alice acting alongside Ben Whishaw and Judi Dench. Alexander also had a supporting role in the final series of Skins, playing Cassie Ainsworth's stalker in the two-part episode "Skins Pure", which aired in July 2013. Also in 2013 Google Analytics released an internal corporate video which features Alexander as a demanding shopper under the sub heading "That is what happens when you save on usability".
Alexander plays one of the main characters in the 2014 musical feature film God Help the Girl, where he also sings and plays guitar. The film was written and directed by Stuart Murdoch, the lead singer of the group Belle and Sebastian, as part of the God Help the Girl project.
Alexander also appeared in The Riot Club, the film adaptation of Laura Wade's stage play Posh alongside British actors Sam Claflin, Max Irons and Douglas Booth. Alexander briefly portrayed the vampire Fenton in the British-American horror series Penny Dreadful in episodes aired in 2014.
In the indie film, Funny Bunny first premièred at SXSW 2015, Alexander plays Titty, a troubled teenager in the comedy drama. The film was directed and written by Alison Bagnall, who directed The Dish & the Spoon, while Alexander was a co-writer together with the other two leads. The film was shown at art houses during the summer and was theatrically released on 13 November 2015.
His band Years & Years formed in 2010, with Alexander joining the band as lead vocalist after guitarist Mikey Goldsworthy heard him singing in the shower and liked his voice. Alexander's voice has been commented on for its unique tone, with the music being described as "Nina Simone ...crooning a melodic lullaby to the folktronic lovechild of Beach House and Terry Riley".
Their debut single proper "Traps" achieved support from Radio 1 and Radio 6, as well as many online publications such as The Guardian and The Fader. "Traps" also features on Kitsuné Maison's 15th compilation.
The band released their second single "Real" on Kitsuné and iTunes in February 2014. The music video featured an appearance from Alexander's Peter and Alice co-star, Ben Whishaw, and former Misfits actor Nathan Stewart-Jarrett. Telling the story in The Independent article of 28 July 2015 of the Ben Whishaw effect (most recent James Bond Q actor), the video received eight and a half million views online and in 2014 they also changed their record label from the French smaller label Kitsuné to Polydor as they gained prominence. In January 2015, they were announced winners of the BBC's Sound of... vote.
Years & Years' most successful single, "King" released under Polydor, headed straight to number one in the UK Single Chart in March 2015, earning the band national recognition. Their debut studio album, Communion, was released on 10 July 2015 and entered the UK Albums Chart at number 1.
As a gay songwriter, Alexander has stated in interviews he would like to see greater use of the qualified male pronoun in music. He explained that, whenever he used to write a diary, he would refer to "you and I" because he wanted to hide who he was writing about. Several Years & Years songs feature male pronouns. "It is kind of sad to me that we don't have gay popstars singing about men using a male pronoun," he told Digital Spy, "but that could change hopefully." Though his work with Years & Years openly references his sexuality, Alexander "can't speak for all gay people, because there are so many different issues, and experiences, and different shapes and sizes. But I can speak for myself, and that is what I'm doing if I'm going to be writing songs and giving interviews, I want to be able to speak about something I care about."
Through various interviews and charity campaigns, he has promoted safer sex and HIV screening anti-LGBT bullying initiatives. He has also spoken openly about his own struggles with depression and anxiety from age 13 onwards. In 2017, Alexander presented a BBC Three documentary investigating the link between being gay and the development of mental health disorders where he opened up about his bulimia.
Alexander describes himself as a "real left-winger" and said in 2016 that he 'had a crush on' Labour Party leader Jeremy Corbyn.