Grant was born Richard Grant Esterhuysen in Mbabane, Swaziland. He adopted the surname Grant when he moved to the UK as an adult and registered with the British Actors' Equity Association.

His father was Henrik Esterhuysen, a man of Afrikaner extraction, who was head of education for the British government administration in the British Protectorate of Swaziland. His mother was a local ballet teacher of South African and German extraction.

As a young child Richard went to primary school at St Marks, a local government school in Mbabane that had only recently become racially integrated. At the age of nine, he witnessed an adulterous relationship between his mother and another man that subsequently led to the break-up of his parents' marriage.

This event stimulated him to keep a daily diary, which he has continued to do ever since. He wears a watch on each wrist, one given to him by his dying father, permanently set on Swaziland time.

Richard went to high school at Waterford Kamhlaba, just outside Mbabane, where he was a day scholar. He studied English and Drama at the University of Cape Town. Richard was a member of The Space theatre company in Cape Town before moving to London in 1982.

"I grew up in Swaziland when it was mired in a 1960s sensibility. The kind of English spoken where I grew up was a period English sound and when I came to England people said 'how strange'. Charles Sturridge, who directed Brideshead Revisited for TV, said 'you speak English like someone from the 1950s'.

He is a successful author, with works such as By Design calling on his Hollywood experience. He is most famous for his role as the drunk Withnail in Withnail and I.

He has twice portrayed the Doctor from Doctor Who, unofficially on both occasions. In the comedy sketch Doctor Who and the Curse of Fatal Death, he portrayed a version of the Tenth Doctor, referred to as the Quite Handsome Doctor. He also voiced a now non-canonical Ninth Doctor for the BBC original animated webcast Scream of the Shalka.