Bossman was born in Accra, the capital of Ghana (then known as the Gold Coast), on 2 November 1955. Bossman came from a relatively well-off family. His father was a politician, as well as a friend and personal physician of Kwame Nkrumah, Ghana's leader from independence. Bossman spent much of his early life in North Africa, Switzerland and the United Kingdom, where his father helped found Ghanaian embassies.
Following a 1966 coup that saw members of the National Liberation Council overthrow Nkrumah's elected government, Bossman was forced to leave Ghana. He was unable to come to Britain, where he had originally hoped to study, and so chose to come to Yugoslavia, originally hoping to study in Belgrade. He was ultimately sent to Ljubljana, however, arriving as a medical student in what was then the Socialist Republic of Slovenia, within Yugoslavia, in 1977. He has said that he "fell in love" with Slovenia when he arrived, finding it "clean and green". While he was studying, he remained politically active, heading an African students' organisation.
His original intention to return to Ghana upon completing his studies was disrupted when he fell in love with a Croatian woman, Karmen Laković, a fellow student. After completing his studies, he and Karmen, also a doctor, settled in Piran, where they run a private practice. They are now married and have two daughters.
Bossman, previously a member of the Piran City Council, successfully ran for mayor in 2010, narrowly defeating his opponent, the incumbent independent centre-right mayor Tomaž Gantar. In the run-off election on 24 October, he achieved 51.4% of the vote, defeating Gantar, who got 48.4%. Unofficial figures following the counting of all ballots showed Bossman with 3418 votes, 185 more votes than the 3233 cast for Gantar. The relatively low turnout, 44% of all voters, was deplored by both candidates.
Bossman's campaign focused on encouraging people to drive electric cars, building a golf course to promote tourism, and promoting local products. He has also pledged to work to try to improve Croatia–Slovenia relations, which have been recently strained over a number of minor border disputes. Bossman has stated that race was not much of an issue during the election, although he did receive criticism on account of his supposed lack of fluency in Slovene. He has stated that a friend, a professor of Slovene, has offered to give him additional lessons.
Bossman officially took office at the first (constitutive) meeting of the municipal council on 12 November 2010. His inauguration as mayor attracted more members of the media, including international media, than there were council members present.
Bossman was re-elected for a second term as mayor on 19 October 2014.
Bossman's election as mayor has attracted international attention. In addition to becoming Slovenia's first black mayor, he has also been described as being probably the first black person to become a mayor in the former Yugoslavia as well as in the former Eastern Bloc (although Slovenia as a part of Yugoslavia was only part of the Eastern Bloc briefly until the Tito–Stalin Split, which led to the eventual formation of the Non-Aligned Movement). He has also been widely compared to U.S. President Barack Obama, being dubbed by the media as the 'Obama of Piran'. Bossman has, however, insisted he has no political ambitions beyond the mayoralty, having stated that he intends to return to medicine upon leaving office.