Julius, the North Fork is special. If you want to DAY hike it, you need a regular Mt Whitney Main Trail Day Hike permit, which means you apply in the lottery for: Mt. Whitney Zone- Day Use All Routes JM34.5MWZ
If you get that permit, you are good to go -- up the MR, down the main trail -- all in one day.

However, if you are overnighting, you need a completely different setup: Search for Inyo National Forest, Permits and Wilderness, click Search. That should get you a screen with two choices (scrolling down) you see:

Inyo National Forest - Wilderness Permits
and
Mt. WHITNEY. Select the Inyo National Forest link. Under the next screen, on the left under Find Permits, you would select "Overnight, Exiting Mt Whitney", and on Entry Trail: "North Fork Lone Pine Creek JM34".

Enter the date and group size, then click Search. On the next screen click See Details (are we having fun yet???). Next screen shows you two sets of choices, first is for:
Select your Entry Location and Entry Date
second is
Select your Exit Location and Exit Date

If you can find an available entry date and exit date combination, you are good to go.
Sure is a pain, but I don't know how they could make it much easier, given all the rules and regulations we have for the Whitney Zone.