As well as this route is the most visited route on Aoraki/Mt. Cook, it is the most dangerous one.
First time this route has been climbind in February, 1912 by Hugh Chambers, Jack Clarke, Jim Murphy and Hugh Wright
The easiest way from Mt.Cook Village is to take a ski-plane to the Grand Plateau. It costs money, but saves your time and forces. Info about this service you can get here http://www.mtcookskiplanes.com/
Another option - from the Mt.Cook village approximately 20-25 km of quite good road which ends in 4 km before the Old Balls Hut - the first shelter on the route. Then follow the track which goes down from the moraine to the Tasman glacier 200 meters from the Hut. Then by the wide moraine pocket head up the Tasman glacier until its junction with Ball Glacier. Then cross the Ball glacier and by sometimes hardly visible trek by slide-rock slopes get to the base of Boys Glacier (be careful of falling stones). Climb the glacier and head up to the Boys Pass. (crevices). From Boys Pass traverse the easy snow slopes to the Cinerama Col. There is avalanche danger on this part of the route. From the Cinerama Col descent to the Grand Plateau and traverse it to the Plateau Hut (a lot of crevices!) From the Balls Shelter 10-12 hours.
There are a lot of crevices in the lower part of the glacier and also it is very high risk of icefalls going down from the right side (Divide).
Start early in the morning while the snow is hard and move as quick as possible in the lower part of the route which is the most dangerous. From the upper part of the Linda Glacier traverse to the left by the Linda Shelf to the Zurbriggen Ridge under the Summit rocks. There is one option to the traversing - a 150 meters gully which goes from the Shelf just to the beginning of the Rocks. Rocks are often climbed by the system of ice gullies which are going down by the rocks. This part is relatively easy but one should climb very carefully, especially when there are people climbing around - the rocks are not stable, a lot of loose stones.
As soon as you are above the rocks, follow the north-east ice cap until the Summit. The easiest way to get to the Summit Ridge is traversing to the West passing the North Ridge and the Sheila Face and then climb the short step and by snow slope get to the Summit.
Also, Linda Route is the most popular descent route - the could be plenty of abseil gear on the Summit Rocks - if you choose to use it for protection, check it twice before using.
The average time of climbing the Linda Route is 15-18 hours, and it is not an easy job to do.
For the route itself: rope, crampones, iceaxes, helmets, 3-4 pitones, 3-4 nuts of middle size, slings, 4-6 ice screws
More info on the Mt.Cook, photos, GPS coordinates you can find on the web page of Alexclimb Mountain School