Bleasdale is an only child; his father worked in a food factory and his mother in a grocery shop. From 1951-57, he went to the St. Aloysius Roman Catholic Infant and Junior Schools in Huyton-with-Roby (then in Lancashire). From 1957-64, he attended the Wade Deacon Grammar School in Widnes (now the Wade Deacon High School). In 1967, he obtained a teaching certificate from the Padgate College of Education in Warrington (which became Warrington Collegiate Institute, now part of the University of Chester).

In 1967, he married Julia Moses; they have two sons and one daughter. For eight years he worked as a teacher at St Columba's Secondary Modern School in Huyton from 1967-71, then King George V School (now The King George V & Elaine Bernacchi School in Bikenibeu in South Tarawa) on the Gilbert and Ellice Islands (now called Kiribati) from 1971-4, and lastly at Halewood Grange Comprehensive School (now known as Halewood College) in Halewood from 1974-5. From 1975 to 1986 he worked as a playwright at the Liverpool Playhouse (becoming associate director) and the Contact Theatre in Manchester (a University of Manchester venue).

His first successes came as the writer of radio dramas for the BBC; several of these plays followed the character of Scully and were broadcast on BBC Radio Merseyside in 1971 (Scully was a young man from Liverpool). Bleasdale's plays typically represented a more realistic, contemporary depiction of life in that city than was usually seen in the media.

The character became so successful that Bleasdale wrote a stage play, two novels and in 1978, a Play for Today ("Scully's New Year's Eve"). That same year, he wrote another single play for the BBC1 anthology series, entitled The Black Stuff. This latter play concerned the story of a group of Liverpudlian tarmac layers (slang: the 'black stuff' of the title meaning 'tarmacadam'). The group are assigned a job laying tarmac on a housing development in Middlesbrough, with their hotel and meals being paid for. However, ambition and greed drives four of the six men to doing a 'foreigner': a job 'on the sly' behind their supervisor's back, laying fresh tarmac for a farmer. However, this turns out to be a catastrophic disaster, after their life's savings are drained on it but end up being swindled by two Irish gypsies. The mental anguish of the four is seen, but they and the others from the collective group are sacked by the project manager.

Although the play remained untransmitted for two years as it waited for an available slot, on its eventual broadcast in 1980 it won much praise, and producer Michael Wearing of BBC English Regions Drama managed to commission the sequel serial that Bleasdale had already been working on. The series, Boys from the Black Stuff, was transmitted on BBC Two in 1982. It established Bleasdale as one of Britain's most important television writers and social commentators.

Bleasdale scripted the film No Surrender (1985), a black comedy in which a group of elderly Protestant hardliners are booked into a party at a pub on the same night as a group of Catholic old-timers.

Since Boys from the Blackstuff, Bleasdale has been responsible for further award-winning television dramas, including The Monocled Mutineer (1986, BBC One), G.B.H. (1991, Channel 4) and Jake's Progress (Channel 4, 1995) the story of a couple (Robert Lindsay and Julie Walters) struggling to cope with a 'difficult' child (Barclay Wright). In 1999 Bleasdale adapted Oliver Twist for ITV. The adaptation was well received but attracted some controversy as Bleasdale expanded the narrative by adding a backstory.

In 1987, Charlottetown Festival director Walter Learning presented the Canadian premiere of the Bleasdale musical Are You Lonesome Tonight? at the Confederation Centre of the Arts, a national arts centre located on Prince Edward Island. A storm of controversy erupted across the country with opponents decrying the presentation of the tough look at the life of Elvis Presley at the festival, which had been known for lightweight family fare such as Anne of Green Gables - The Musical. The chairman of the Board of Directors resigned in protest over the Bleasdale play and objections to the rough language and subject matter were even brought up in the provincial legislature. However, the play was allowed to proceed and became a major critical and financial success for the festival that season.

He is currently writing a two-part serial about the World War II ocean liner RMS Laconia which will be broadcast on BBC Two in 2010. The story revolves around the lives of the people on board the liner and their hardships during the sinking.