Burkinshaw began his footballing career with Midland League side Denaby United while working at Dodworth Colliery. He had a brief spell as an amateur with Wolverhampton Wanderers before joining Liverpool in November 1953. He played just once for Liverpool, against Port Vale in April 1955, moving to Workington in December 1957 for a fee of £3,000. He was player-manager of Workington between November 1964 and March 1965, leaving to join Scunthorpe United in May 1965, having played 293 league games for Workington. He played a further 108 league games for Scunthorpe, and had a short spell as caretaker manager before retiring from playing in May 1968.
Shortly after announcing his retirement, Burkinshaw moved to Zambia where he coached for a few months before returning to England as coach of Newcastle United. He was sacked by Newcastle in May 1975 and joined Tottenham Hotspur as coach the same month.
He acted as manager of Tottenham HotspurFootball club from 14 July 1976 to 31 May 1984. Where he won more major football competitions than all but one other Spurs manager (that being Bill Nicholson). Spurs were relegated in Burkinshaw's first year in charge but bounced straight back for promotion the following year. He signed two ArgentineWorld Cup stars, Osvaldo Ardiles and Ricardo Villa, in 1978. It was considered a brave move but Ardiles would become one of the Spurs greats and Villa would score one of the greatest goals ever seen at Wembley in the 1981 FA Cup Final replay. Burkinshaw's Spurs, with Ardiles, Villa and Glenn Hoddle, won two successive FA Cups (81 & 82).
In his final game in charge, Spurs won the UEFA Cup (1984) for a second time after a penalty shoot-out after the second leg at White Hart Lane. In doing so, they beat an R.S.C. Anderlecht team that included the future Spurs Sporting Director Frank Arnesen. On leaving White Hart Lane for the last time, brought about by a disagreement with the board, he remarked "There used to be a football club there".
In June 1984 he was appointed as coach to the Bahrain national side. He left that role in July 1986. Burkinshaw later managed Sporting Clube de Portugal until he was sacked in February 1988. In October 1988 he returned to England as manager of Gillingham, but resigned in April 1989 with the team on the verge of relegation to Division Four.
Burkinshaw was later Chief Scout for Glenn Hoddle and Ossie Ardiles at Swindon Town and in May 1992 became assistant to Ardiles at West Bromwich Albion. When Ardiles moved to manage Tottenham in the summer of 1993, Burkinshaw was promoted to Albion manager. However, his career as Albion manager lasted just one season (1993–94) and he was sacked after they narrowly avoided relegation back to Division Two.
He was later Director of Football at Aberdeen before briefly taking charge as caretaker-manager at Pittodrie when Roy Aitken was sacked in 1997, he left when Alex Miller was appointed as the club's new manager.
In March 2005 Burkinshaw was appointed assistant manager at Watford. He left this position in December 2007 due to a serious family illness, having helped the club reach promotion to the Premier League in 2006.