Kazinsky was born Robert John Appleby in Haywards Heath, the son of Phyllis and Paul Appleby. He grew up in Brighton. He has an older brother, Michael. Kazinsky is Jewish, his parents are descended from immigrants from Poland and Russia. He is a fluent speaker of Hebrew. Kazinsky was educated at Hove Park School in Hove, from 1995 to 2000. He appeared in their productions of Bugsy Malone and A Midsummer Night's Dream. Kazinsky trained as an actor at the Guildford School of Acting from September 2002 to July 2005. He took his grandfather's middle name as his stage name.
Kazinsky appeared in Israeli television commercials. He made his acting debut in 2005, when he had a guest role in an episode of the CBBC children's programme The Basil Brush Show, playing a character named Sven Garley. Kazinsky then got a role in Sky One's football drama TV series Dream Team, in which he played Casper Rose from late 2005 to 2006, when he was killed off towards the end of the ninth series after the actor announced his intention to leave.
On 18 May 2006, it was announced that Kazinsky would be playing Sean Slater in EastEnders, and he made his first appearance on 22 August 2006 and last appearance on 1 January 2009.
On 27 March 2007, Kazinsky was suspended from work on EastEnders for two months. A story in The People newspaper alleged that Kazinsky had bombarded a model with "obscene text messages and photographs." Kazinsky issued a statement apologising "for any offence caused and for bringing the show into disrepute".
On 18 January 2008, it was announced that Kazinsky would leave EastEnders. Speaking to This Morning on 31 July 2008, Kazinsky confirmed that he would not be pursuing a career in Hollywood, but would instead begin an apprenticeship as a mechanic when he leaves EastEnders.
On 22 October 2010, it was announced that Kazinsky had been cast in the film The Hobbit: An Unexpected Journey. He would have played Fili, a member of the Company of Dwarves, but in April 2011, Kazinsky resigned from the role due to personal reasons.
On 9 January 2011, he guest starred on the U.S. drama TV series Brothers & Sisters as Dr Rick Appleton.
On 13 November 2012, it was announced he would join the cast of the sixth season of the U.S. fantasy drama TV series True Blood in 2013. In 2013, Kazinsky appeared in the Guillermo del Toro film Pacific Rim, as Chuck Hansen, an Australian Jaeger pilot.
In 2015, he portrayed Guy in the science fiction film Siren. In March 2015, he was cast in the leading role in the U.S. Fox drama series Second Chance. His character, James "Jimmy" Pritchard, was a murdered elderly ex-cop who is genetically brought back to life as a man in his mid-30s with unusual powers. The series debuted in early 2016 to somewhat disappointing ratings.
Kazinsky co-starred as Orgrim Doomhammer in the Warcraft film adaptation, which was released in May 2016.
