Greig was born to a Scottish immigrant father and a South African-born mother, and was educated at Queen's College, Queenstown, South Africa. Many former Sussex players had been recruited to coach the cricket team at Queen's College—during Greig's schooldays, Jack Oakes, Alan Oakman, Ian Thomson, Ron Bell, Richard Langridge and Mike Buss all came from overseas for off-season work. All of them noticed Greig's developing abilities which, after a first-class debut for Border Province in the Currie Cup, led to a trial at Sussex when Greig was 19. Greig's father helped him decide between university study or pursuit of the Sussex offer. "He used to slam into me for not reading enough, for being generally immature. He would look at me sometimes and say 'Boy, when I was your age I was fighting a war', but in the end he grinned and said: 'Go over to England for one year, one year mind, and see what you can do'".

After Greig scored 156 in 230 minutes against a strong Lancashire attack in his first game for Sussex, his future direction changed irrevocably. He wrote a brief note to his father, telling him he would not be coming back to go to university. Greig set a goal of making the England Test team in six years, though he returned to play in South Africa during the winter for a number of years, eventually transferring to Eastern Province for the 1970–71 season.

Greig was now experimenting with finger-spin to complement his medium pace. He set off to the West Indies with the England team in early 1974 and ran straight into a major controversy. On the second day of the First Test at the Queen's Park Oval in Trinidad, the West Indies had cruised to a first innings lead of 143, thanks mainly to 142 not out from Alvin Kallicharran. With four wickets still in hand, the home team was in a dominant position when the last ball of the d