Caspi was born in 1949 in Kibbutz Hanita in the Upper Galilee, and is of Romanian ancestry on his grandfather's side. He studied piano at the conservatory in Nahariya.
After his military service he married Galia Superstein, and in less than a year they divorced. In 1972 he met actress (Patty) Doreen Lubetzky, and three years later they got married. They had two children, Brit (born 1981) and Bar (born 1985).
In 1990, following marital difficulties, Caspi separated from his wife and met Raquel Wenger. The new couple emigrated to Canada. The two married in 1994 in California. They had two children, Suyan (born 1992) and Sean (born 1995). Caspi was granted a divorce by the Beverly Hills Rabbinical Court, although at the time the case was still pending in Israel. In 1997 Caspi returned to Israel and appeared in several concerts, including the Arad festival. In 2002, a Tel Aviv court determined that he was still legally married to Doreen at the time he married Raquel in 1994, and found him guilty of bigamy, and imposed a six months suspended sentence and a small fine. An appeal to the District Court failed, and in 2004 the original sentence was upheld.