The route to Empress Hut is highly technical and only suitable for very experienced mountaineers.
The approach and conditions change constantly through the year, depending on the snowpack. There are multiple hazards.
Mountaineering routes to/from Empress Hut are included in Aoraki Tai Poutini: A guide for mountaineers by Rob Frost. It is available for sale at the Aoraki/Mount Cook National Park Visitor Centre or on the New Zealand Alpine Club website.
There is also information available on ClimbNZ about the Hooker Glacier area.