Alexander was born in Oxford and was a chorister at Magdalen College. He studied conducting and singing at the Royal Academy of Music and was a piano pupil of Michael Dussek. He then went on to study musicology and analysis at Cambridge University with a choral scholarship from Selwyn College. After graduating in 2004, he was immediately appointed as répétiteur and conductor at the National Opera Studio in London, during which time he received many awards and prizes. He was music director of the Stadttheater Klagenfurt from the 2013/14 to the 2015/16 seasons.

Notable engagements elsewhere: Bayerische Staatsoper (La bohème) and he will make his debut at the Wiener Staatsoper (Il barbiere di Siviglia), the Dresden Semperoper (Der Freischütz), and the Metropolitan Opera in New York (La bohème). In Mannheim he will lead, among others, productions of Parsifal, Fidelio, Rosenkavalier, Aida and Tannhäuser.