Accam did not belong to a club during the first twelve years of his life. He instead spent his time playing with friends on gravel pitches in the parks of Accra. There he was spotted by a scout from Ghana-based Right to Dream Academy where he would spend four years.
In 2008 Accam received a three-year scholarship to study at Hartpury College in England. He combined his time there with playing in the English lower leagues. After signing for English club Ledbury Town, Accam scored on his debut in December 2009. He then played for English club Evesham United during the 2010–11 season while also featuring in the reserve team squad at Forest Green Rovers. The fact that Accam was in England on a student visa prevented him from playing for their first team, or any club in a higher division.
After his time at Evesham United he took part in the Nike "The Chance" competition where he was one of eight winners who got to spend the 2011–2012 season with Nike Football Academy.
In March 2012 Accam signed for Swedish third tier side Östersund. English manager Graham Potter was in charge of the club and he had already signed two other Ghanaians who were studying with Accam at Hartpury College. And after scouting him at the Nike Academy he brought Accam with him back to Sweden. Accam became an immediate success with his new team and several Allsvenskan clubs became interested in signing him for the summer transfer window.
Accam was then signed by Allsvenskan club Helsingborg in August 2012, and was due to be loaned back to Östersund for the remainder of the season, but was recalled by Helsingborg for an upcoming Champions League qualifying match against Celtic. The 2 million SEK transfer fee paid by Helsingborg was the highest ever for a player from the Swedish third tier.
After scoring seventeen goals in twenty-six appearances, the club announced that Accam would be leaving Helsingborg to pursue opportunities elsewhere. After he made the announcement, Accam was linked with a move away from Tigres, Eintracht Frankfurt and VfB Stuttgart, a number of English clubs and CSKA Moscow. Accam rejected a move to CSKA Moscow, having insisted that Accam hadn't been in talks with the club.
Despite initially denying the move, Accam signed as a designated player with Chicago Fire of Major League Soccer on 19 December 2014.
Accam received his first call-up to the Ghana national team for a 2014 FIFA World Cup qualification match against Sudan on 24 March 2013, in which he was an unused substitute.
On 12 May 2014, Accam was named in a provisional 26-man squad in the 2014 FIFA World Cup by Ghana manager James Kwesi Appiah. However, on 2 June 2014, Accam was left out of the World Cup squad. Accam expressed his disappointment of being dropped for the World Cup squad.
On 15 November 2014, Accam made his international debut for Ghana in a 1–0 defeat to Uganda in a 2015 African Cup of Nations qualifier, coming on as a 67th-minute substitute for Christian Atsu.
He was selected as part of Ghana's preliminary squad for the 2015 Africa Cup of Nations.Scores and results list Ghana's goal tally first.
Accam received his U.S. green card in October 2015 which qualifies him as a domestic player for MLS roster purposes.