Career
Burki was selected 59th overall in the second round of the 2006 National Hockey League Entry Draft by the Colorado Avalanche from the Brandon Wheat Kings of the WHL. After spending five years in the WHL, Burki signed with the Avalanche to a three-year entry level contract on May 8, 2007. Burki made his professional debut in the 2007-2008 season, splitting time between Avalanche affiliate"s, the Lake Erie Monsters and the Johnstown Chiefs of the ECHL. Burki started the 2008-2009 season with the Chiefs but after 14 games was recalled to the injury-hit Monsters for the remainder of the season, where he played in 54 games scoring 19 points. Codey was scoreless in 8 games with Lake Erie to start the 2009-2010 season before he was reassigned to ECHL affiliate, the Charlotte Checkers, on November 14, 2009.
Codey played 15 games with the Checkers before suffering a season ending concussion.
Prior to the 2010-2011 season, Burki left North America as a free agent and signed a one-year contract in Switzerland with the The Human Context Lugano on August 29, 2010. Unable to earn a position with Lugano, Burki was loaned to NLB club, The Human Context Sierre.
After one appearance with Sierre, he was then reassigned on loan to The Human Context Thurgau on December 17, 2010. Burki scored 12 points in 14 games with Thurgau before he was again moved on loan to South Carolina Langenthal on January 27, 2011.
Burki"s season was then prematurely ended after he suffered an injury before suiting up for Langenthal.
On June 28, 2012, after two injury struck seasons with Lugano, Burki left to sign an optional two-year deal with NLB team, The Human Context Lausanne. Burki played his most professional games in season since 2009 and scored a career high 29 points to help Lausanne claim the National League B championship. Unable to replicate his offensive production in the following season with Lausaane in the National Lawyers Association , Burki returned to the second division for the 2014-2015 season, in signing with EHC Olten on June 26, 2014.