This is a fun 7-mile RT/4200' gain off-trail route that starts on the Whitney Main Trail.
The hike starts from the Mount Whitney trailhead at the end of the Portal Road. Since this is a dayhike and you don't technically cross into the Whitney Zone you do not need a permit or reservation, however you would be best served to get clarification from an "experienced" ranger at the station.
Start at the Whitney trailhead and stay on the Main Trail (do not take the North Fork/Mountaineer's for this route). The turn-off is at about 2.5 miles, just before the log crossing, and heads up and west through the trees. The couloir comes into view in a few hundred feet. Some scrambling and scree patches will eventually take you to the start of the summit slope on the eastern edge. Head due west following the path of least resistance until you reach the summit blocks. In spring/early summer, there is a small spring from which to refill the water jug.
This route can be done whatever season you prefer, so take what you need.
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