While the North Window probably isn't the most popular arch to photograph in Arches National Park, (Delicate Arch takes that honor) it's one of the best. The North Window perfectly frames a view of Turret Arch and the expansive red rock landscape westward towards Canyonlands National Park.
The east side of the arch catches morning sun, with the richest colors ocuring right at sunrise. Turret Arch, seen through the window, is touched by light right before the North Window, and the distant canyons a moment before that. Like Mesa Arch, being set up and ready before the sun rises is the key to catching the best light. The North Window is a short, half mile walk from the car, making it an easy access, early morning photo. Here's a link to the exact time of sunrise.
You'll want a tri-pod for tack-sharp images in the low light. A wide angle lens is necessary to frame the window and take in the whole scene. Also consider adding a polarizing filter to your lens, it will make the skies in your photo a deeper blue and add to richness in the color of the rock. Here's the settings from my camera for the photo above: ISO 400, shutter speed: 1/180 of a second, f-stop f6.3.
There's a small elevated ledge east of the North Window where it's easy to line up Turret Arch through the "window frame." On weekend mornings don't be suprised to find other photographers there with you. After all, this is one of the iconic images of Utah.
Getting there: From Moab, drive north on Hwy. 191 for five miles. Turn right into Arches National Park and continue 9.2 miles to the turn-off for the Windows Section. Turn right just past Balanced Rock and continue to the end of the road and the trailhead for The Windows and Turret Arch. Hike less than half a mile to the North Window, walk through the window to east side and scramble up to a small ledge to shoot through the North Window frame.