Boysen joined AB as a youth player from Herlev IF. He was invited on a trial at Italian club Reggina in December 2007 at the age of 16. Boysen have also bin on trials at Premier League clubs Liverpool and Chelsea. In February 2012, Boysen signed his first contract with AB until the summer 2011. He was promoted to the first team in the summer 2008. He was a huge talent and got already his debut at the age of 17, in a match against Kolding FC in the Danish 1st Division, where he replaced Henrik Lyngsoe Olsen in the 85th minute on 10 August 2008. Already in his third game for the first team, he scored a goal against FC Roskilde in the Danish Cup on 24 August 2008. Boysen was nominated for the Talent of the Year 2008 by the Danish Football Association. Boysen played 24 league games and scored 8 goals in his first season despite his young age, and was wanted by clubs from both the Netherlands and Germany.
After a one-week trial at Celtic in January 2010, he was linked with a move to the Scottish club. He played a friendly match for Celtic, and scored after five minuted. However, his agent later revealed, that Boysen wasn't in top form due to the football winter break in Denmark, and therefore didn't offer him a contract but they would like to follow him and see him again, when he was in top form.
After a good half season in the 2010/11 season, with 16 league games, 4 goals and 4 assist's, he was sold to Lyngby Boldklub. He played in total 63 games for the first team.
Lyngby announced on 31 January 2011, that they had signed Boysen from AB valid from the summer 2011. But the following day Lyngby confirmed, that Boysen instead would join the club with immediate effect. Boysen played his first game for Lyngby on 14 March 2011 against OB, which Lyngby lost 0-2. He scored his first goal for the club already in his second game against AaB.
In 2012, Israeli club Hapoel Tel Aviv attempted to bring Boysen and Swiss defender, Guillaume Katz, in on loan as the club was trying to strengthen their squad targeting Jewish footballers. He went on a trial at the club but stayed at Lyngby for now.
After three good seasons at Lyngby, Boysen was followed by Danish Superliga clubs. According to some medias, Boysen was in negotiations with Esbjerg fB. Lyngby offered him a contract extension, but he rejected it and announced on 8 January 2014, that he would be leaving the club immediately.
Boysen signed with Viborg FF on 13 January 2014 on a free transfer. He played his first game for the club on 23 February against Randers FC. It was head coach Ove Christensen that wanted to transfer Boysen to Viborg, but after Ove Christensen left the club and was replaced by Auri Skarbalius, Boysen found it hard to become a regular part of the squad, after he played 7 games from start and one from the bench, without scoring any goals. After a disappointing first half season, he left the club.
Lyngby announced on 27 June 2014, that they had signed Boysen back on a two-year contract. Boysen's first season back at Lyngby, was an extraordinary season. He was named as the profile of the year in the Danish 1st Division after only 6 months in the league. He scored 12 goals in 24 league games before getting injured for the rest of the season in May 2015.
In the next season, he became the profile of the year in the Danish 1st Division once again. With 18 league games and 11 goals, he was sold in January 2016.
Boysen signed with Brøndby IF on 25 January 2016 and was reunited with his former manager from Viborg FF, Auri Skarbalius. He got his first game for Brøndby on 28 February against Hobro IK. Boysen was in good form during the spring, and this aroused interest from foreign clubs.
On 8 July 2016 after only 6 months in Brøndby IF, Boysen was sold to Dutch club Roda JC. In December 2016 Boysen revealed, that he wasn't happy at the club due to the teams results which affected his stay in the Netherlands.
After 13 league games and 1 goal, Boysen's contract was terminated.
Despite the interest from Beitar Jerusalem and Hammarby IF, Boysen signed for Lyngby for the third time in February 2017. After his transfer to Lyngby he revealed that he had rejected foreign clubs that offered him millions in sign fee.
Boysen's father is from Israel.