Grant was born at Charing Cross Hospital in Hammersmith, London, the second son of Fynvola Susan MacLean (b. Wickham, Hampshire, 11 October 1933; d. Hounslow, London, July 2001) and Captain James Murray Grant (b. 1929). His grandfather, Colonel James Murray Grant, DSO was decorated for bravery and leadership at Saint-Valery-en-Caux during World War II. Genealogist Antony Adolph has described Grant's family history as "a colourful Anglo-Scottish tapestry of warriors, empire-builders and aristocracy". His ancestors include William Drummond, 4th Viscount Strathallan, Dr. James Stewart, John Murray, 1st Marquess of Atholl, Heneage Finch, 1st Earl of Nottingham, Sir Evan Nepean, and a sister of former Prime Minister Spencer Perceval.
Grant's father was an officer in the Seaforth Highlanders for eight years in Malaya and Germany. He ran a carpet firm, pursued hobbies such as golf and painting watercolours, and raised his family in Chiswick, west London, where the Grants lived next to Arlington Park Mansions on Sutton Lane. In September 2006, a collection of Capt. Grant's paintings was hosted by the John Martin Gallery in a charity exhibition, organised by his son, called "James Grant: 30 Years of Watercolours". His mother worked as a schoolteacher and taught Latin, French and music for more than 30 years in the state schools of west London. She died at the age of 67, 18 months after being diagnosed with pancreatic cancer.
On Inside the Actors Studio in 2002, he credited his mother with "any acting genes that [he] might have". Both his parents were children of military families, but, despite his parents' backgrounds, he has stated that his family was not always affluent while he was growing up. He spent his childhood summers shooting and hunting with his grandfather in Scotland. Grant has an older brother, James, living in Portugal.
Grant started his education at Hogarth Primary School in Chiswick but then moved to St Peter's Primary School in Hammersmith; he was then educated at an independent preparatory school, Wetherby School. From 1969 to 1978, he attended the independent Latymer Upper School in Hammersmith on a scholarship and played 1st XV rugby, cricket and football for the school. He also represented Latymer on the quiz show, Top of the Form, an academic competition between two teams of four secondary school students each.
In 1979, he won the Galsworthy scholarship to New College, Oxford, where he starred in his first film, Privileged, produced by the Oxford University Film Foundation. He read English and graduated with 2:1 honours. Actress Anna Chancellor, who met Grant while she was still at university, has recalled, "I first met Hugh at a party at Oxford. There was something magical about him. He was a star even then, without having done anything."
He received an offer from the Courtauld Institute of Art, University of London to pursue a PhD in the history of art, but decided not to take the offer because he failed to secure a grant. Viewing acting as nothing more than a creative outlet, he joined the Oxford University Dramatic Society and starred in a successful touring production of Twelfth Night.