Born in Downpatrick, Deeny was educated at St. Patrick's Boy's Primary School and St. Patrick's De La Salle Grammar School, both in Downpatrick. During this time he regularly partipated in several sports, representing Ulster Schools at Table Tennis, representing County Down in the Gaelic Athletic Association and playing football in both the Irish League and League of Ireland in the early 1970s.
He studied at University College Dublin and worked as a general practitioner in County Tyrone from the mid 1980s onwards. From 2000 he also served as Chairman of Omagh and District G.P. Association and took a prominent role in the campaign to keep full medical provision at the Tyrone County Hospital.
In the 2003 Northern Ireland assembly elections Deeny ran as an independent candidate in West Tyrone on the sole issue of retaining the hospital and generated one of the biggest shocks of that election when he topped the poll and took a seat from the Social Democratic and Labour Party. However the continued suspension of the Assembly has meant that Deeny has not been able to directly influence decisions about the future of the hospital.
In the 2005 general election Deeny stood for the Westminster seat, campaigning heavily against sitting MP Pat Doherty's abstentionism and arguing that this denied the seat a voice, and received a lot of backing from many activists and supporters of both Nationalist and Unionist political parties, though all the major parties ran candidates. Deeny received over 11,000 votes and placed second in the election to Pat Doherty. In the 2007 Assembly elections Deeny was elected on the seventh count with 3,776 first preference votes. His campaign was again largely based on a single issue - the impending closure of the Sion Mills branch surgery.
In the first two months since the restoration of the Northern Ireland Assembly, Doctor Deeny has claimed £13,863.37 in expenses (which averages out to approx. £1732.92 per week over 8 weeks) and is the second highest expenses claim for an MLA.