Alexandrou was born in Hackney, London, to a Greek Cypriot father and an English mother. He has two sisters, one older, one a twin, as well as a younger brother.
He was educated at Chingford Foundation School in Waltham Forest, as well as at Anna Scher Theatre School. Other notable former pupils of Chingford Foundation School include footballer David Beckham, and Sir Jonathan Ive, who is the chief design officer (CDO) of Apple Inc.
Alexandrou's acting career began in 1996, when he auditioned successfully for the role of Martin Fowler in the BBC One soap opera EastEnders. The character was created shortly after the show's inception in 1985, with the role becoming vacant following the departure of actor Jon Peyton Price in 1996. Alexandrou appeared as Grant in the short film Blessed Burden in 1999.
In July 2003 he participated in a "Soap Stars Special" edition of The Weakest Link. Over Christmas of 2005, Alexandrou starred in the crime caper, Who Stole The Snowman? On 14 April 2006 it was announced that Alexandrou would leave EastEnders in Autumn 2007. He and Natalie Cassidy, who played his on-screen wife Sonia Fowler, left on 2 February 2007 (to tie in with the exits of Wendy Richard and Ray Brooks). When asked about his departure Alexandrou said, "I've decided to leave the show to experience other aspects of my industry. Having turned 21, I felt it was a good time to leave. I owe everything to the show and would love to come back one day - that's if EastEnders would have me." On 12 October 2014 it was announced that actor James Bye would be taking over playing the role in the character's return to Albert Square.
In early 2007 Alexandrou appeared in the play The Homecoming opposite Harold Pinter for BBC Radio 3.
Alexandrou toured the UK and Norway with the British Shakespeare Company from June–September 2007. He played Pistol in Henry V, and Orlando in As You Like It. It was incorrectly reported that he turned down the chance to replace Daniel Radcliffe in the West End revival of Equus in 2007.
In 2008 he appeared in In My Name, a new play by Steven Hevey, at the Old Red Lion Theatre produced by Yaller Skunk Theatre Company. The play transferred to the Trafalgar Studio 2 from 1–19 July 2008. In October 2008 Alexandrou appeared in All Quiet on the Western Front at the Nottingham Playhouse prior to a UK tour of the production and therefore was unable to attend the wedding of former EastEnders actor Wendy Richard and John Burns.
In March 2009 he played the part of Romeo in the Globe Theatre's production of Romeo and Juliet. In October 2009, Alexandrou joined up with EastEnders cast members, Natalie Cassidy (Sonia Jackson) and Nina Wadia (Zainab Masood), to star in comedy shorts for BBC Raw Words.
In 2010 Alexandrou filmed his first feature film, Semper Fidel in Cuba. The film was released in 2014. He appeared in the role of The Man in John Goodrum's play The Black Veil a thriller adapted from a Charles Dickens short story.
In 2011 Alexandrou took part in the BBC Learning project, Off By Heart Shakespeare, where he played Mercutio from Romeo and Juliet delivering the speech; "'tis not so deep as a well".
In 2012 Alexandrou toured theatres throughout the United Kingdom, with the Hull Truck Theatre starring as Phil, in the play DNA.
In April 2014, as part of the Shakespeare 450 celebrations, he played the classic villain Iago in Grassroots Shakespeare London's production of Othello at Leicester Square Theatre in London's West End and also performed sonnets at the Guildhall with Damian Lewis. Rehearsals were recorded by BBC World Service and James Alexandrou's performance as Iago featured in their programme The Why Factor.
In February 2015, Alexandrou appeared in Silent Witness as Carl Parry.
Alexandrou was in a relationship with Kara Tointon which began in April 2006. They split on 4 November 2007.