Born in Sydney, New South Wales, Klemmer played his junior football for All Saints Toongabbie before being signed by the Parramatta Eels he also played Australian rules football and was once playing for the Sydney Swans academy. At 15 years of age, Klemmer signed with the Canterbury-Bankstown Bulldogs. In 2010 and 2011, Klemmer played for the Australian Schoolboys. In 2011, Klemmer also played for the New South Wales Under 18s team. He played for the Bulldogs' NYC team in 2011 and 2012, scoring 6 tries in 28 games. In 2012, Klemmer was named on the interchange bench in the inaugural Under 20s State of Origin match for New South Wales. At the end of 2012, Klemmer was named at starting prop in the 2012 NYC Team of the Year. He was also named NYC Player of the Year. On 5 March 2013, Klemmer re-signed with the Bulldogs on a 3-year contract.
In Round 1 of the 2013 NRL season Klemmer made his NRL debut for the Bulldogs against the North Queensland Cowboys. Also in 2013, Klemmer was named captain of the New South Wales team in the Under 20s State of Origin match, leading the team to a victory over Queensland. On 5 October 2014, in the Bulldogs 2014 NRL Grand Final match against the South Sydney Rabbitohs, Klemmer played off the bench in the Bulldogs 30-6 loss.