Gaup was born in Kautokeino, Finnmark County in Northern Norway. He first intended to become an athlete but from 1974 to 1978 he went to drama school and studied at the Beaivváš Sámi Theatre in Kautokeino. He also founded the first Sami language theatre ensemble.
After acting in several movies, he rose to international prominence in 1987 with his film Ofelaš (international English title Pathfinder). It was the first full-length movie with all of the dialogue in Northern Sámi. This movie earned him an Academy Award nomination for Best Foreign language film and the Grand Prize award at the 1990 Yubari International Fantastic Film Festival. After that he made the Disney-financed movie Haakon Haakonsen (Shipwrecked), based on the youth adventure novel Haakon Haakonsen. En norsk Robinson (Haakon Haakonsen. A Norwegian Robinson) by Norwegian author O. V. Falck-Ytter. In 1993 he shot his most successful film to date Hodet over vannet (Head Above Water). For this film, he won the Amanda Award (the most important Norwegian Film Award). In 1996 it was remade as an American movie with Cameron Diaz and Harvey Keitel in the leading roles. His next movie was Tashunga (also known as North Star), a project by Christopher Lambert. This film was panned by both the box office and the critics.
Gaup was previously scheduled as director for the Kevin Costner movie Waterworld. But due to exploding costs, he left the project.
In 2008 he made Kautokeino-opprøret which is about the Kautokeino rebellion of 1852 of the town with the same name in Norway. It is the true story of the riots of the Sami people against the church and state domination on alcohol sale.
Awards and nominations
-Amanda Award Winner - Pathfinder
-Academy Award Nomination Best Foreign language film - Pathfinder
-Amanda Award Winner - Head above Water