Follow the maintained trail from the parking lot for 6 miles to Sam Mack Meadow. Cross the stream to the east side, and look for an unmaintained trail that heads up and east. It is about 50 yards south of the meadow's north end. If you find yourself scrambling about on a disappearing path, you've probably not found the correct path. Follow the very good trail for two miles until it peters out on the east side of the Palisade Glacier's lateral morraine. From here, the route goes up the canyon over granite benches and ledges, following a series of cairns. Stay on solid rock as far as possible before cresting the ridge.
You now have two choices: Head directly for the U-Notch (and losing some elevation enroute) or contour to the left on Mt. Gayley's west side to avoid losing altitude. Both routes are used, pick your own poison. Personally, I dislike contouring as it stresses the ankles. Don crampons when you reach the glacier, and head for the U-Notch.
The U Notch Couloir is Class 3. It is the unmistakeable couloir between the summits of Polemonium on the left and North Palisade on the right, when viewed from the north. It is 700 vertical feet, at 40 degrees. If the bergshrund is open (as is likely in August), it is best crossed on the rocks on the far right side. This can be the most difficult rock climbing of the entire route, and goes 5.4-5.6, depending on conditions. Secor calls this an easy class 5 section, but I found it a bit tougher than that. Although it is short (~25 feet), you should plan to use a rope in this section unless you are very comfortable on such terrain.
Once on the snow/ice in the couloir, you can climb free (with axe and crampons) if the snow is sufficiently soft. If hard, you may want to use a rope even though this will slow progess considerably. The bergshrund is generally wide enough to swallow you at any speed should you fall and fail to arrest yourself. It is dark and scary inside, and can ruin your whole day. Do not underestimate the ammount of nerve this requires for those unfamiliar with such couloir climbing.
Two thirds of the way up, the couloir is split by a rocky peninsula. Pass to the left where the snow reaches higher towards the U-Notch. This eventually leads to easier rock and scree which can be followed to the notch. From the U-Notch, you can choose either the class 4 Clyde variation or the 5.4 Chimney variation. The Chimney route is obvious and easy to follow. The same cannot be said for the Clyde route, which involves climbing down 100 feet on the south side of the U-Notch before ascending a series of class 4 sections. Again, choose your own route here. The Chimney route is one very full pitch. It is not a chimney in the usual sense of the word, as it is large enough to drive a truck through once you squeeze through the first narrow passage about 10 feet from the start. There are numerous slings along the way that can be used to clip into, but none of them will help you in a fall (they are for rappelling, mostly). Climb as high as your rope will allow, reaching a small ledge just below the crest of the ridge. You should see slings and several pitons in various places that can used to set an anchor.
From here, it is a mostly class 3 scramble to the summit. There are some very large summit blocks that must be surmounted just before the summit, but none of them require the use of a rope.