Kluane First Nation has the right to hunt and trap within Kluane National Park, in accordance with their Final Agreement:
http://www.ainc-inac.gc.ca/pr/agr/klu/fia2_e.pdf

Also see the Kluane National Park and Reserve Management Plan: http://www.pc.gc.ca/pn-np/yt/kluane/images/kluane_mgmt_plan-en.pdf. Page 13 addresses the issue of KFN and CAFN's hunting rights within park boundaries. From that document:

"When the National Parks Act was amended in 1976 to include Kluane National Park Reserve, reserve status was applied only until an aboriginal land claim was settled in the region. After this amendment First Nation people were allowed to hunt and fish in the park reserve, but fearing reprisals, few did. Many former Southern Tutchone trails and campsites have fallen into disuse. The 1993 Champagne and Aishihik First Nations Final Agreement reaffirmed CAFN traditional gathering, hunting, fishing and trapping rights in the newly designated Kluane National Park. Although these traditional rights have been constitutionally guaranteed with the Final Agreement, four generations of Champagne and Aishihik First Nations members have seen their intimate ties with the park eroded and eventually effaced. Since the early 1940s, the same cycle has taken place for the Kluane First Nation. Renewing an intimate relationship with plant and animal communities within the park and traditional cultural sites and resources is a high priority for both the Champagne and Aishihik First Nations and Kluane First Nation. This cultural reintegration will also help ensure the continuity and vibrancy of the aboriginal traditional knowledge of this region."