Amundsen was born in Harstad to construction and real estate chief Per Roald Amundsen (born 1949) and health secretary Wenche Berit Trudvang (born 1949).
He attended Kanebogen elementary school, finishing in 1984, before enrolling in Harstad Secondary school, graduating in 1987. He later went to Heggen Upper Secondary School between 1987 and 1990. Upon finishing school, he was drafted to the Norwegian Army, serving his tour in the Signal Battalion in Bardufoss. Upon being discharged, he enrolled at Norwegian School of Economics, graduating in 1995 with a degree in economics. In 1999, he received a degree in information technology at the Sør-Trøndelag University College. He also completed the Military leadership course with the Home Guard.
Amundsen joined the Progress Party's Youth when he was fourteen years old, and was the chairman of its Harstad chapter from 1986 to 1990 – the last year also of its Troms chapter. From 1992 to 1994 he served as board member of the Hordaland Progress Party Youth, and from 2000 to 2002 he was chairman of the Harstad Progress Party.
From 1999 to 2005, Amundsen held various positions in the Harstad municipality council. Since 2005, he has been a Member of Parliament from Troms. During his time in parliament, he has been a member of the Standing Committee on Local Government and Public Administration. In 2011, Amundsen changed from being the party's Spokesperson on Immigration Issues, to Spokesperson on Energy Issues. After the 2013 parliamentary election, Amundsen, located 2nd on the party list, lost his seat due to re-districting of Troms county. He was subsequently appointed state secretary in the Ministry of Local Government and Modernisation in the new government of Erna Solberg.
On 20 December 2016 Amundsen was appointed Minister of Justice, succeeding fellow Progress Party politician Anders Anundsen.
Before the 2005 election, he stated as his wishes to work in the background within the Standing Committee on Business and Industry or the Standing Committee on Energy and the Environment. By the latter part of his 2005–2009 term as Member of Parliament, he however rather came to be mostly notable for his views and proposals on immigration issues. While he formerly favoured Norwegian membership of the European Union, he has since the late 2000s turned to oppose membership. He has stated that his political rolemodel is Margaret Thatcher.
Amundsen has said that his concern with immigration is largely due to the problem of unintegrated immigrants. He says that many adhere to cultural values which "collide fundamentally" with Norwegian and Western values such as democracy, freedom, freedom of speech, equality and tolerance. He does not accept that all cultures are equally good or valuable. He views radical Islam as "fascism based on religion", which he puts in his context of communism as "fascism based on understanding of class" and nazism as "fascism based on race". He has said that he believes radical Islam should be fought against like all types of fascism.
His political proposals include to establish asylum centres in areas close to where the asylum seekers originate from, and in general to drastically reduce immigration to Norway. He has also sought to remove the public economical support for the Islamic Council Norway as it did not distance itself from the use of death penalty for homosexuality. He wants stricter control on Mosques, not accept halal slaughtering, remove halal food from prison facilities, and reject hijab in the police force. He also oppose special demands such as separate schools for Muslims, classrooms segregated by gender, limitation of freedom of speech regarding religion, special breaks for praying during work-time and alcohol-free days in nightclubs. He claims that the political left, socialists and politically correct people are giving in to many such demands.
He supported the proposal of a ban on burqa and niqab in public spaces which was proposed by his party in 2010. He stated that "we don't accept this type of clothing [which is] oppressive to women, and what they represent". The same year, Amundsen also proposed to put the immigration policy up for a referendum, and proposed a halt of immigration to Oslo in the wake of increased segregation and soon-to-become immigrant majority areas, which he said would turn integration impossible.
Amundsen supports oil exploration off the Lofoten archipelago, a major breeding ground for Atlantic cod. He also supports the involvement of Statoil, the Norwegian oil and gas company where the state of Norway is the majority shareholder, in exploitation of oil sands.
Amundsen defines himself as climate sceptic. He has argued that climate problems are used to conduct socialist policies, saying that "for many socialists, CO2 has replaced Karl Marx".
He was married on 11 July 2009 to Gry-Anette Rekanes-Amundsen, herself a local Telemark Progress Party politician. They divide their time between Oslo and their rural estate in Flåbygd in Telemark. In addition, they own a house together in Harstad.