Bonnet was born in Skegness in 1947. He had his first hit single with the duo, The Marbles in 1968, with the single "Only One Woman" (Polydor 56 272), which reached Number 5 in the UK Singles Chart. This and its follow up were both written by Barry Gibb and Robin Gibb of the Bee Gees who had recorded in Australia with Bonnet bandmate from The Marbles, Trevor Gordon.

Bonnet then quit to do advert jingles. He appeared in the 1974 British comedy film, Three for All as lead singer of 'Billy Beethoven', a fictional band, along with several notable UK comedy personalities and his then partner Adrienne Posta, although his character's lines were limited to only two words.

In 1977 he released an eponymous album, which was certified gold in Australia. The single, "It's All Over Now, Baby Blue", a cover version of the Bob Dylan song, also reached the top five in Australia in 1977, and the following year the single "Warm Ride", written by the Bee Gees, a leftover from the Saturday Night Fever sessions, reached number one there.

In 1979, Bonnet was chosen by Ritchie Blackmore to replace Ronnie James Dio as the vocalist of hard rock band Rainbow. This was something of a musical departure for Bonnet, who had previously identified himself more as an R&B singer. Bonnet would later credit his time in Rainbow, and his collaboration with Blackmore in particular as changing fundamentally his musical outlook to a more hard rock focus. He sang on the Down to Earth LP, which would become his most successful album. It spawned two hit singles in 1980: "Since You Been Gone" and "All Night Long". During Bonnet's time in the band, Rainbow also headlined the inaugural Monsters of Rock festival at Donington Park, Castle Donington.

Bonnet's time with Rainbow was short and he left to resume his solo career, releasing the Line-Up album in 1981, handled by producer John Eden. Following on from his time in Rainbow, the album had a markedly more rock-based sound than his previous solo recordings, whilst retaining some of his former R&B influences. For the recording of Line Up Bonnet enlisted several well-known rock musicians including Whitesnake guitarist Micky Moody, Whitesnake and Rainbow drummer Cozy Powell, Deep Purple and Whitesnake keyboard player Jon Lord, and Status Quo guitarists Francis Rossi and Rick Parfitt. The album reached #62 in the UK Albums Chart. The album's lead single, "Night Games", reached #6 in the UK Singles Chart, with the follow-up single, "Liar", reaching #51. Around this time Bonnet sang on an advertisement for Levi's jeans (the song was entitled "These Eyes"), although his version has never been released.

Tempted by an offer from ex-UFO guitarist Michael Schenker, Bonnet continued his progression to a heavier musical style and joined the heavy metal Michael Schenker Group (MSG) for the Assault Attack album.

Upon leaving MSG, Bonnet decided to form his own band, drawing inspiration from his days in Rainbow. In 1983, he co-founded Alcatrazz, with Gary Shea (bass), and Jimmy Waldo (keyboards) of the band New England, former Alice Cooper drummer Jan Uvena, and Swedish guitar virtuoso Yngwie Malmsteen. The band's debut album, No Parole from Rock N' Roll, featured a distinctly heavy metal sound, with Bonnet describing the material as the heaviest that he had recorded to date. The band enjoyed moderate success in the United States, and major success in Japan.

The band released three studio albums (No Parole from Rock N' Roll, Disturbing the Peace, with guitarist Steve Vai replacing Malmsteen, and Dangerous Games with Danny Johnson on guitar). Alcatrazz also released a live album, Live Sentence, with material from Yngwie Malmsteen's period in the band. For the Dangerous Games LP, Alcatrazz recorded a remake of Bonnet's hit song "Only One Woman" from his first band The Marbles. In 1987, Alcatrazz appeared on the second series of the BBC TV programme Rock School, with Bonnet providing tips of singing. The programme also included clips of Alcatrazz in rehearsal. Later that year, Alcatrazz disbanded, and Bonnet returned to his solo career. In 2009, Bonnet reformed Alcatrazz with three new musicians, and has begun recording new tracks for a new Alcatrazz studio album scheduled for release in 2010. Bonnet did not consult with his former bandmates Shea, Uvena, and Waldo about re-forming the original line-up, and in response they also re-grouped as a separate version of Alcatrazz. Both bands have official pages on MySpace.com.

Bonnet's post-Alcatrazz projects have all been short-lived and included Blackthorne (working with ex-Balance guitarist Bob Kulick, brother of Kiss guitarist Bruce Kulick), backing vocals for the Danish heavy metal band Pretty Maid's Future World and numerous session work for Forcefield, appearing on To Oz and Back and Let The Wild Run Free. In 1988, Bonnet joined Impellitteri, for their album Stand in Line.

An appearance on Eddie Hardin and Pete York's musical adaptation of Wind in The Willows, performed live in Germany in 1991, saw Bonnet sing on around six of the songs alongside fellow musicians Jon Lord, Don Airey and Ray Fenwick amongst others. Fenwick and Airey also featured heavily on Bonnet's 1991 solo album Here Comes The Night which includes several covers as well as songs written by his wife Jo Eime, and another remake of the Marbles', "Only One Woman".