Birkin has published photo-essays and articles in Frieze, Cabinet, Creative Time Reports, Ibraaz, Disegno, The Harvard Advocate, and the American Civil Liberties Union blog on subjects ranging from a legally protected species of iguana roaming freely at Guantanamo Bay detention camp to Marilyn Monroe's 1945 photoshoot at an army drone factory in California.

Birkin has also worked as an actor in film and theatre, including productions for the Royal National Theatre, Peter Hall Company, and Brooklyn Academy of Music, and narrated the English translation of the 1953 film by Chris Marker and Alain Resnais, Les statues meurent aussi, at the French Institute in London. He appeared in two episodes of Star Trek: The Next Generation: in "Family" as René Picard, Captain Jean-Luc Picard's nephew; and in "Rascals" as a young Jean-Luc Picard. His film credits have included roles in The Return of the Musketeers (1989, as Louis XIV), Impromptu (1991), Les Misérables (1998), All the Queen's Men (2001), Charlotte Gray (2001), and Sylvia (2003). He has since performed in films by the Artist Nathaniel Mellors for the 2009 Tate Triennial, the 2010 British Art Show at the Hayward Gallery, the 2011 Venice Biennale, and Ourhouse at the Institute of Contemporary Arts in London.

Birkin grew up between London and New York City, eventually returning to the UK to study human sciences and anthropology at Oxford University (1996–1999). He completed an MA at the Slade School of Fine Art (2009–2011) with a scholarship from the Arts and Humanities Research Council.

Birkin has exhibited internationally, including at The Courtauld Institute of Art, The Photographers' Gallery, Saatchi Gallery, Trolley Books, and Photo50, London; Baibakov Art Projects, Moscow; Photomonth, Kraków; the Benaki Museum, Athens; Centre d'art et de photographie de Lectoure, France; Mudam Museum of Modern Art, Luxembourg; FotoFest, Houston; MoMA PS1 Rockaway Dome, and the Whitney Museum ISP in New York. He was the recipient of the 2010 Sovereign Art Prize (Barbican Centre, London), the 2012 Celeste Prize for Photography (Museo Centrale Montemartini, Rome), and a photography bursary from the National Media Museum.

Birkin was a fellow of the Art and Law Program in New York (2011–2012) and the Independent Study Program at the Whitney Museum of American Art (2012–2013). He was an artist-in-residence at Yaddo in 2013, the MacDowell Colony in 2015, and Lower Manhattan Cultural Council's studios on Governors Island in 2016.

His first public solo exhibition, "Mouths at the Invisible Event", opened at The Mosaic Rooms, London, in 2015. Hyperallergic described it as "a methodical examination of the language, aesthetics, and ethos of modern warfare [that] ultimately makes the emotional reality and Kafkaesque lunacy of such a system hit home."