Goddard was born in Croydon, London, England, in 1962. Between 1974 and 1979, he attended Ravensbourne School, Hayes Lane, Bromley, London. A punk in 1977, he helped form a band at school called The Belsen Horrors (the name of which came from a Sex Pistols track). First a Drummer, then the lead singer, he and his band lasted until 1978. Later, sometime between 1979 and 1980, he was the Drummer in another Bromley punk band called The Vamp. This group was led by Wayne Cregan, who was one of the first members of X-Ray Spex. Bromley was closely linked with the early punk movement – first through the Bromley Contingent (the first fans of the Sex Pistols), and three famous punk Singers who came from this town: Poly Styrene (X-Ray Spex), Siouxsie Sioux (of Siouxsie and the Banshees) and Billy Idol – who like Trevor, attended Ravensbourne School for Boys.

He appeared in cameo roles in many television shows prior to 1995. That same year, Goddard played Kano in the film adaptation of Mortal Kombat. His performance as Kano became the source of the character's evolution in the video games.

He would go on to act in other films such as Gone in Sixty Seconds, Men of War (with Dolph Lundgren and Future JAG co-star Catherine Bell) and Hollywood Vampyr. He played a short-tempered mercenary in the 1998 film "Deep Rising" with Treat Williams and Famke Janssen. Goddard played in the television drama series JAG as Lieutenant Commander Mick Brumby and was featured in TV commercials for the Hoover FloorMate.

On 7 June 2003, Goddard was found dead in his home in North Hollywood, Los Angeles, California. He was 40 years old. Initial reports indicated that Goddard was in the process of getting a divorce and that suicide was suspected. An autopsy later showed that Goddard died from a drug overdose of heroin, cocaine, temazepam and vicodin.