Coxhead graduated from the University of Waikato in 1994 with a Bachelor of Laws.[1] He worked in private practice before becoming a lecturer at the University of Waikato Law School.[1] He served as president of Te Huinga Roia Māori o Aotearoa – the New Zealand Māori Law Society.[1]
Coxhead was appointed to the Māori Land Court on 25 January 2008. In 2011 he was appointed as a judge of the High Court of Niue, and in 2016 as a Justice of the High Court of the Cook Islands.[1] He was appointed Chief Justice of Niue on 22 November 2018, replacing Patrick Savage.[2]