Born in Guyana, Grant settled in the United Kingdom with his family in 1960. His brother, Eddy Grant, was a member of The Equals in the late 1960s, and Rudolph also recorded in this era, as Little Brother Grant, releasing the single "Let's Do It Tonight". In the 1970s, he recorded as a reggae deejay under the name The Mexicano, releasing singles such as "Gorilla in Manilla", and "Cut Throat". Grant had his biggest success in 1978, with his Starsky & Hutch-inspired single "Move Up Starsky", on which he deejayed over Bob Marley's "I'm Still Waiting" rhythm. The single topped the UK reggae chart, with an album of the same name following it, and inspired the less successful "Move Up Hutch" by Superstar. Grant followed this with several other singles, including "Lover's Conversation", but none matched the success of "Move Up Starsky".
By 1981, Grant had abandoned the Mexicano alias and began recording as a singer under his own name, recording cover versions of songs including John Lennon's "Woman", and, most successfully Stevie Wonder's "Lately", with which he had a minor UK Singles Chart hit in February that year. The single's success led to a contract with Stiff Records, although he failed to score any further hits.
Grant's 1987 single "Mash in Guyana" proved a major success in his country of birth, and has been described as "the unofficial anthem of Mashramani". He wrote the song on a visit to Guyana in 1986 and recorded it at his brother Eddy's Coach House Studios on his return to London. Grant performed the song at the Notting Hill Carnival and it went on to top the first soca chart in London, published by the Black Echoes music paper.
In 2002, Grant's version of "Lately" was included on the Dynamite Reggae Classics album.