Kemp was born in Hertfordshire, England, the son of Rosi, a former model, and Barry Kemp, a graphic designer. Trained at the Royal Ballet School, at age 17, he was accepted at dance company, Adventures in Motion Pictures (AMP). He won the lead role of The Swan in the all-male cast of Matthew Bourne's Swan Lake from 1997 to 2000 in both London and Broadway. In 2002 he could be seen dancing to "Stuff Like That" in a Peter Lindbergh-directed commercial for Gap's "For Every Generation" campaign and in 2004, he appeared alongside Sarah Jessica Parker in two Francis Lawrence-directed spots, "Color" and "Shine", for Gap's "How Do You Share It" campaign. He famously turned down a lucrative modelling contract with Giorgio Armani, citing not wanting to be confused as a model as opposed to an actor as the reason for his decision.
Kemp made his acting debut in the 2004 film Van Helsing, opposite Hugh Jackman and Kate Beckinsale. He played the horse Nugget as well as The Young Horseman in the 2007 West End revival of Equus starring Daniel Radcliffe. The part of Nugget required him to don a wire horse mask and metal hooves/horseshoes, and carry Radcliffe on his back whilst galloping. He had a role in Step Up 2: The Streets as Blake Collins, director of the Maryland School of the Arts (MSA) and brother of Chase Collins.
Kemp and his wife, Gaby Jamieson, have a daughter, Thalie.