Born in Glostrup, Schöne started playing youth football with Himmelev-Veddelev BK and moved on to B70 and then Lyngby BK. In 2002, he moved to the youth team of Dutch Eredivisie team SC Heerenveen. He spent four years with the team without playing any league games, and moved to Eerste Divisie team De Graafschap in 2006. In his first season, he helped De Graafschap win promotion for the Eredivisie. In his two seasons with De Graafschap, he scored 12 goals in 70 games for the club.
Schöne signed on with NEC Nijmegen in the summer of 2008. He scored the winning goal for NEC away at Spartak Moscow in December 2008, which sent the team through to next round of the UEFA Cup. The club and the fans picked him as "Player of the year" in NEC, a title he also won at his previous club Graafschap.
On 18 April 2012, Schöne signed on with AFC Ajax. The contract is valid until 2015. Schöne is the 21st Danish player to play for Ajax, and one of his predecessors has been a role model for him through his career. "Now I will be playing for the same club as Michael Laudrup did. I modelled myself on him."
Schöne began his international career with the Denmark under-16 national team, scoring a goal in his debut in August 2001. He played a total 10 games and scored one goal for the Danish under-17 and under-18 national teams from July 2002 to March 2004. After he joined De Graafschap, Schöne was once more a part of the Danish youth squads, as he made his debut for the Denmark under-21 national team in March 2007. He played a total nine games and scored one goal for the under-21s until October 2008.
On 12 August 2009 Schöne made a stunning debut for the Danish national team in a friendly against Chile, when he scored less than a minute after being put into action in the second half.