Alexander is a Professor at the Te Piringa Faculty of Law, University of Waikato. He obtained his LLB and LLM degrees with Honours from The University of Auckland. He did his PhD at Nottingham and post-doctoral studies at Columbia University in New York City. His areas of scholarship pertain to international and comparative environmental law; the laws of war; civil liberties; and a number of pressing issues of social concern.

He has published nineteen books. His most recent works are People, Power and Law: A New Zealand History (jointly written with Professor Claire Breen, which was published in 2022 by Bloomsbury/Hart in Oxford, UK); and Volume IV of his Causes of War (1650-1800) series. This was also published by Bloomsbury, in 2021. He has also written over forty academic articles.

Alexander has been awarded a Rotary International Scholarship, a Fulbright Fellowship, a residency at the Rockerfeller Bellagio Centre in Italy, and the New Zealand Law Foundation International Research Fellowship. He was the recipient of a Francqui Foundation award with which he held a professorship at Ghent University, Belgium, for six months during 2018/2019. In 2021 he was the joint winner of the Critic and Conscience of Society Award.

He has also been the lawyer/expert on a number of international delegations and advised the New Zealand government on multiple matters of international concern. Professor Gillespie was the first New Zealander to be named Rapporteur for the World Heritage Convention, involving international environmental diplomacy under the auspice of UNESCO. Alexander has also been engaged in policy formation for the United Nations, the Commonwealth Secretariat, and governmental, commercial and non-governmental organisations in New Zealand, Australia, United States, United Kingdom, Ireland and Switzerland. He has also made a number of appearances before the Waitangi Tribunal and Select Committees of the New Zealand Parliament.