[6][7] He attended St Kilian's German School in Clonskeagh and later studied for the Leaving Certificate at the Institute of Education, before going on to graduate from University College Dublin with a degree in law in 2004.

In 2011, Murphy participated in Freedom Flotilla II, which attempted to breach Israel's blockade of the Gaza Strip.

[21] In 2012, Murphy set up the website ScamBridge.org for people wishing to share their experiences of the government's JobBridge scheme.

[23] In 2013 while discussing the Israel-Palestine conflict, Murphy argued that "Palestinians have a right to defend themselves against that aggression and that may involve armed defence against soldiers".

He said "people wouldn't be bothered" and defended the move as a response to Fianna Fáil candidate Mary Fitzpatrick, who had also put up her posters early.

On 30 September 2019, Murphy announced a new political group called RISE (Radical Internationalist Socialist Environmentalist).

In April 2022, he defended the decision of his and fellow People Before Profit TDs not applauding Volodymyr Zelenskyy's address to the Dáil.

[34] In September 2022, Murphy was assaulted by protesters outside Leinster House while leaving the Dáíl,[35] and in April 2023 his family home was targeted by protestors.

As a TD for the Anti-Austerity Alliance and member of the Socialist Party, Murphy took a leading role in the We Won't Pay campaign, an anti-water charges organisation advocating non-payment.

[42] On 9 February 2015, he was arrested by Gardaí at his home, in relation to the Jobstown protest, and taken into custody along with three other anti-austerity activists.

[46] On 15 September 2015, Murphy was served a summons by the Gardaí on charges of false imprisonment of Joan Burton and her assistant.