Born in Amsterdam, Biseswar's father is a bus driver in Amsterdam. Growing up, his only passion was football.
Biseswar's first football club was amateur club DWS. He joined Ajax after a year. After five seasons with Ajax, he was deemed "unmanageable" and was released. Bisewar has stated that he was happy to leave Ajax, saying that "Ajax thought I wasn't good enough and I wanted to leave myself. I didn't feel at ease, many friends were leaving and my relationship with the coach wasn't optimal. It was unfortunate, but there were no dreams in tatters."
Biseswar returned to former club DWS, before leaving again to play for Telstar. After one season, Biseswar was asked to join the Feyenoord youth academy, a move he was happy to make: "I visited them a few times and was quickly convinced. I had no difficulty to join Feyenoord as a boy from Amsterdam. And why? Ajax was far behind me and I always had a special feeling for Feyenoord."
After playing for Feyenoord U15 for one season, Biseswar was promoted to Feyenoord U17, in which he played for two seasons. At the end of the last season, Biseswar made his debut with Feyenoord's first team.
On 29 October 2004, he signed his first professional three-year contract with Feyenoord, at the age of 16. On 22 May 2005, Biseswar made his senior Feyenoord debut by replacing Tim Vincken in the 67th minute in Feyenoord's last Eredivisie match of the 2004–05 season. Feyenoord defeated ADO Den Haag (6–3), which was also the last match under the management of Ruud Gullit.
In the 2005–06 season, Biseswar mainly played for Feyenoord U19. However, Biseswar also made six appearances in Feyenoord's first team and scored his first goal in the Eredivisie. On 20 November 2005, Biseswar replaced the injured Ali Boussaboun in the 53rd minute in the Eredivisie away match against Roda JC. In the 79th minute, Biseswar scored the winning goal.
Biseswar had two Eredivisie loan spells while at Feyenoord. For the 2006–07 season he joined Heracles Almelo. Biseswar made 26 Eredivisie appearances for the club, in which he scored two goals. However, Heracles Almelo decided not to retain Biseswar for disciplinary reasons.
For the 2007–08 season, Biseswar had an unsuccessful spell at Eredivisie side De Graafschap. Biseswar failed to make an impression and only played 11 Eredivisie matches for the club from Doetinchem, without scoring any goals.
For the 2008–09 season, Biseswar returned to Feyenoord; his contract was due to expire at the end of the season, and he was placed on the transfer list. He trialled at English club Sheffield United, but they could not afford the player. Feyenoord failed to sell the youngster, but Biseswar made a good impression during pre-season and the early league matches. Therefore, manager Gertjan Verbeek opted to give Biseswar another chance at the club.
Over the course of the season, Biseswar became an important first team member. As his contract was due to expire, he received interest from former club Ajax and Spanish club Valencia. On 10 March 2009, Biseswar announced he didn't want to leave Feyenoord for another Dutch club. On 3 April 2009, Biseswar signed a new three-year contract with the club. Biseswar finished the season with 30 Eredivisie matches, scoring a total of seven goals.
Biseswar moved to Turkish club Kayserispor in January 2012, signing a four-and-a-half-year contract. Until the expiration of his contract in summer 2016 he made 130 appearances scoring 19 goals in all competitions.
In June 2016, Biseswar signed a three-year contract with Greek club PAOK on a free transfer. On 6 May 2017, the Dutch winger along with Pedro Henrique Konzen bagged the goals for Serbian manager Vladan Ivic's team, helping PAOK win the Greek Cup final with score of 2-1 against AEK Athens at Panthessaliko Stadium of Volos. This marked 5th time in the club's history that the Greek Cup Title was captured. Biseswar finished the 2016-17 season with 7 goals and 10 assists in 44 appearances in all competitions.
Biseswar has played for the Netherlands at under-16, under-17, under-19, under-21 and B-international level.
Biseswar played at the 2005 UEFA European Under-17 Football Championship in Italy, scoring a late group stage winner against Israel. The goal took the Netherlands through to the semi-final against hosts Italy. Biseswar was mentioned in the team of the tournament, but couldn't save his team from losing in the final to Turkey. With a second place, the Netherlands U17 qualified for the 2005 FIFA U-17 World Championship in Peru. Biseswar's team finished third after losing the semi final against later winner Mexico U17 (0–4), but winning the third place play-off match against Turkey U17 (2–1).
Biseswar's parents are from Suriname, although he was once (incorrectly) though to be eligible for Trinidad and Tobago citizenship. He has one older sister and one younger brother, who is named Romario. The Biseswar boys were named after football icons Maradona and Romário.
Despite growing up in Amsterdam, Bisewar's family are fans of the Rotterdam-based Feyenoord club.