Greenwich was born in New Zealand to a Georgian father and American mother. His father, Victor Greenwich Dadianov (formerly the Honorary Consul-General of Georgia in Sydney, 2004–2013), was born Prince Victor Dadianov of the princely Georgian Dadiani family ^{[citation needed]} but his mother changed the name to Greenwich after they moved as refugees to New Zealand from Russia after the Second World War.^[2] At the age of seven, Greenwich moved with his family to Sydney. From his family residence in Circular Quay, Greenwich was educated at Sydney Grammar School and completed a Bachelor of Arts in Human Resource Management and Russian Studies at the University of New South Wales.^[3] From 1 December 1998 to 1 December 2012, Greenwich was the Managing Director of his own recruiting agency, Winning Attitudes Recruitment.^[4]

He is a direct descendant of the House of Dadiani, one of the oldest royal houses in Eastern Europe.^{[5] [citation needed]}

Greenwich is the only openly gay male MP in the NSW Legislative Assembly. Before entering politics, Greenwich was a prominent LGBT rights activist and led Australian Marriage Equality (AME).^[6]

Prior to running for office, Greenwich was the national convener of AME from 2009, and in 2010 was named as one of Samesame.com.au's 25 most influential gay and lesbian Australians.^[7] As national convener, Greenwich organised over 44,000 submissions to be made to the 2011 Australian Senate inquiry into same-sex marriage, and continues to be a prominent activist for achieving same-sex marriage reform in Australia.^[7] In May 2012, Greenwich married his German Australian long-term partner, Victor Hoeld, in Argentina, where same-sex marriage was already legal.^[8]

As of 2017 Greenwich has triple Australian, New Zealand and United States citizenship.^[9]

In 2023, Greenwich was one of those honoured as part of the 'Pride 2023' campaign. During this Sydney's Queen Victoria Building (QVB) underwent a transformation, including a temporary renaming, celebrating Sydney WorldPride. The QVB paid tribute to five members of the LGBTQIA+ community by capturing them as ‘Real Queens’ in a series of royal–style portraits.^[10] For WorldPride, Greenwich swapped his signature casual suit and shirt for a tuxedo with green tulle in a display that was hosted in the QVB.^[11]

In May 2023, Greenwich announced he would be launching defamation action against Mark Latham for his homophobic tweet about Greenwich. Greenwich has also made a formal complaint to police against Latham, for using a carriage service to harass and offend, and has also lodged a complaint of homosexual vilification to the anti-discrimination board.^[12]

In July 2012, Greenwich aligned himself with prominent independent Lord Mayor of Sydney Clover Moore and announced his candidacy on Moore's ticket for the Sydney City Council elections scheduled for the NSW local government elections in September 2012.^[13] This low (and therefore most likely unelectable) position on the ticket fuelled speculation that this was to increase Greenwich's visibility for a possible run to succeed Moore should she be forced to resign her state seat of Sydney in light of promised reforms by the O'Farrell Liberal/National government to ban MPs from serving on local government bodies.^[14] This legislation was subsequently passed as the Local Government Amendment (Members of Parliament) Act, 2012 (NSW)^[15] and following the local government elections in which Moore was re-elected for a third term as lord mayor, Moore resigned her seat in the New South Wales Legislative Assembly, triggering a by-election.^[16]

Greenwich subsequently contested the 2012 Sydney by-election as an independent with the endorsement of Moore, comfortably defeating Shayne Mallard of the Liberal Party with a 47.3 percent primary and 63.7 percent two-candidate preferred vote.^[17] Greenwich said after the by-election that it was "very clear Barry O'Farrell's legislation has backfired – because now there are two of us".^[18] Greenwich has denied claims that he is a single-issue politician,^[19] having gone to the by-election on a platform involving a range of policy areas, including small business, the re-establishment of an inner-city public high school, and social welfare and public housing, among others.^[20]

Greenwich introduced the Reproductive Health Care Reform Bill 2019 into the New South Wales Legislative Assembly on 1 August 2019, in a bid to decriminalise abortion in New South Wales, allow abortions for up to 22 weeks, and permit an abortion after 22 weeks if two medical practitioners agree. The bill passed the parliament on 26 September and was given royal assent on 2 October 2019 as the Abortion Law Reform Act 2019.^[21]

• Alex Greenwich – Independent for Sydney
• Greenwich v Latham Federal Court of Australia online file