In 1971 Graham spent four months on the dole and attended 30 job interviews, at the 31st interview he was offered a position with Winthorp Pharmaceuticals which introduced him to the opportunity of forming Galpharm International in 1982. Galpharm began life in a shed on Firth Street in Huddersfield and rapidly expanded, relocating first to Brighouse and then to purpose built premises in Dodworth on the outskirts of Barnsley. The business was innovative and creative in being a pioneer in lifting restrictions on which pharmaceuticals could be sold making them more freely available and accessible both in pharmacy and grocery outlets. This ensured several hundreds of millions of pounds of savings per annum to the NHS in both time and cost. Under Leslie's control the company became the UK's largest licensed supplier of OTC and Healthcare medicines for both branded & Private Label customers and the largest in the world when it was subsequently acquired by the Perrigo Group in 2008 for $88m.
In 2006, he was awarded an Honorary Doctorate of the University of Huddersfield and six year later he became Visiting Professor of Enterprise and Entrepreneurship, and has a base at the University’s 3M Buckley Innovation Centre.
On the 17th June 2017 Graham was honoured in the Queens Birthday Honours and awarded a C.B.E. by His Majesty Prince William during a ceremony at Buckingham Palace.