Start at Mount Cook Village. Go to Hooker Glacier and then the Gardiner Hut. Then climb to the Empress Hut.
The climb is brilliant. From Empress Hut climb the northwest butress or the couloir. Climb to Low Peak. Now follow the grand traverse to High Peak. It is a difficult climb and many have died on the traverse. From High Peak you can climb down the normal route on the Tasman Glacier. The weather is very bad on Aoraki Cook so you must take at least a week supply. Ten day supply is better.
This is a roped climb so take all the expedition ice gear.
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