He was one of three brothers who played instruments with Leone Band Club of Victoria. As a singer, Dominic took part in local festivals and his first vocal group was The Rosebells with his brother Alfred and a couple of friends living in the same area. Though the group was a vocal one, there were moments when both Dominic and Alfred played their band club instruments with the group. Then, while still a student, Dominic formed a beat group by the name of Gemini 5. Then he founded The Tramps in Victoria, Gozo, in the mid-sixties managed by George Said known as Id-Dias. The pop group made its first studio recordings in 1974 with a line-up comprising Grech (lead vocals and guitar), Joe Cassar (keyboards and vocals), Spiro Sillato (vocals and bass guitar) and Carmel Portelli (vocals, drums and percussions).
The Tramps were undoubtedly the most popular Gozitan pop group ever. Their most successful recordings include Xemx and Inti Djamant. The former remained popular for decades and was even played and sung by the Malta Philarmonic Orchestra and Ivan Grech at the annual Rockestra with a musical arrangement by Josef Debrincat. Inti Djamant is not officially Gozo's anthem, but most Gozitans treat it as much. In the eighties The Tramps won the Festival Internazzjonali tal-Kanzunetta Maltija 1981 with Sinjur Ġib il-Paċi, a song which was even played and sung on the occasion of Pope Benedict XVI's visit to Malta in 2010. The band which accompanied the choir included Richard and Mark, two of Dominic's 4 sons.
He was married to Pauline nee Ellis. They have four sons: George, Richard, Christian and Mark.
The Tramps made their last public appearance in December 2004, where they gave their traditional Christmas gig in Victoria. They had been doing this for about twenty five years. Dominic played for the last time at the Nazarene church in Sliema, together with his sons Richard and Mark during a prayer meeting.
He died at the age of 55 years at the Gozo General Hospital after a prolonged illness.