Graham was born in Cheadle, just south of Manchester. As a youth he was on the books of Manchester United F.C. as a part of their youth program. Graham gave up football in favour of cycling. He took his first step towards becoming a professional cyclist in 1972, age 14 he joined the Abbotsford Park Road cycling club which then had a clubroom in Fallowfield, Manchester. During his first year of racing he had no success and was even lapped frequently during the circuit races. It was not until 1974 at 16 years of age he won his first race in Macclesfield. He was coached by Malcolm Firth, who, along with Harold "H" Nelson, developed a training routine based on heart rates and power training. In 1975, as a Junior, he gained three fourth places in national championships (pursuit, road race and 25-mile time-trial) – indeed, the 40 km/25-mile time trial was his first race over the distance for two years and covered the distance in a little over 57 minutes. As well as winning the divisional junior road title, he won the senior pursuit title against the established senior rider, Malcolm Fraser. In his last year as a junior he had 12 wins and also finished 22nd in the World Junior Road Race Championship. In 1976 he went to the Netherlands for five months: his best place was 2nd and he had a host of top six places.
In 1977 he won the UK's first classic of the season – the GP of Essex. He then moved to Belgium, along with the Liverpool rider Nigel Hartle. He won at Anvaing. His form saw him selected for the Peace Race, his first international stage race. The season ended with him being selected to ride in the Etoile Des Espiors (Stars of the future) race in France. He raced against both amateurs and professional riders including Bernard Hinault, yet proved he could hold his own against anyone, finishing third in a stage. In 1978 he moved to France to ride for the notable amateur team ACBB (Athletic Club de Boulogne Billencourt), Europe's most successful sports club, following in the footsteps of his friend and training partner Paul Sherwen. He rode in about 65 races, winning 15 of them including the Grand Prix de Nations, Paris-Évreux and the GP de France. International selection included the five-day Sealink International: he won the prologue in his native Manchester. His success in France saw him win the season long Merlin Plage Palme d'Or as the top amateur in France, and this led to him being offered a professional contract with Peugeot.
Graham was one of four new signings among the twenty two riders in the Peugeot team of 1979 which included Bernard Thévenet and Hennie Kuiper. In his first Professional race the Grand Prix St Raphael, he finished in second place behind Roger Rogiers. A few weeks later, on stage 4 of the Paris Nice he was a part of an elite breakaway which tore the race apart. His undoubted class, however, wasn't allowed to flourish as it might, as he was over-raced by his team. Over the coming seasons, his ability against the watch, in the mountains and as a rouleur won him the respect of the peloton.
Throughout his professional career Graham competed in the majority of prestigious races from the Tour de France to Paris–Roubaix. However a combination of his loyalty to the team and bad luck restricted his palmares, as he was often used as a domestique to assist French riders who were behind him overall on general classification. His biggest disappointment was during the 1981 Critérium International, he had dropped Bernard Hinault but on the penultimate descent was knocked off by a press motor cycle. His 1982 season started badly: a training crash caused by ice, whilst out in Cheshire, led to a broken femur and this impacted massively on the season ahead. However, in 1982 he still managed arguably his greatest performance finishing second in the Flanders Classics Het Volk. He caught and dropped the breakaway group that included Sean Kelly and Roger De Vlaeminck. However he did not manage to catch the eventual winner Alfons De Wolf, finishing 26 seconds behind. In 1985 he returned home to Britain, riding with John Herety in the British-based professional Ever Ready /Marlboro team. One year later, Graham moved to the British ANC-Halfords team, riding alongside Joey McLoughlin, Paul Watson and Malcolm Elliott. ANC-Halfords were managed by Phil Griffiths and became a surprise inclusion in the 1987 Tour de France. The team suffered from a shortage of funding, and riders not knowing how to work as a team in a major Tour: as the grandee of the team, Graham tried to bring a professional approach to both the management and the riders Unfortunately, Graham was too exhausted to finish his final Tour. He then had a few rides the following season for Emmelle-MBK riding alongside Mark Bell. However his full-time career as a professional rider effectively ended with ANC-Halfords.
Since his retirement from the peloton, Graham has worked for a bike company (Emmelle) and still works as a commentator for the BBC, Eurosport and ITV. He's a regular on the Tour de France, and gets to meet up with the former riders of the peloton, notably Jean-René Bernaudeau, his former team mate and good friend.