Birch started out as a footballer and was on the books of Liverpool - he was the last player signed by Bill Shankly - but only played for the reserves and by 1979 had left the club altogether. He also had brief spells with Shrewsbury Town and Chester City, after which he retired from playing professionally and became a chartered accountant, playing non-league football with clubs including Marine, Runcorn, Northwich Victoria and Skelmersdale United. Birch failed to score a league goal in his season with Chester but did find the net in the FA Cup against Barnsley.
Specialising in insolvency, Birch became chief executive of Chelsea, a club with a debt burden of around £80m, in February 2002, and attempted to restructure their finances. He also helped put together the £140m Roman Abramovich takeover during June and July 2003. The takeover wiped out most of the club's debts, making Birch's role no longer necessary, and he was replaced by Peter Kenyon. Birch was offered another role, but instead resigned.
In October 2003, Birch was appointed chief executive of Leeds United, another club with high debts, and was briefly chairman following Professor John McKenzie's resignation. He helped the club avoid administration, and oversaw the takeover of Leeds by Gerald Krasner's consortium, before moving on to Everton in June 2004 to become their chief executive. Birch was given the task of overhauling the Merseyside club's finances, but resigned just six weeks later.
In June 2007 he became Chief Executive Officer of Derby County, replacing Mike Horton's previous role as managing director. Horton remained on the board as vice chairman.
In October 2007 Birch left Derby County. He was lined up to be the new Southampton Chairman heading the new regime if the proposed takeover from Daniel Williams had materialised.
On the 20th November 2009, Sheffield United announced Birch as their new Chief Executive, replacing Terry Robinson.