I can't add much to what Steve has already posted, but I have done that route. I did a 10-hour Carillon-Russell-Muir-Whitney trip in September, 2002. I did Muir before Whitney, so I could descend the Mountaineers' Route instead of the trail.
It's been almost ten years, but I'm going to guess that my route up the north face wasn't too different from the red line in Kent's photo. There wasn't very much snow left, but the snow that remained was very icy, so I had to avoid it with light hiking boots. The icy snow and some less-than-perfect route finding meant that I had to do a few short sections of fourth class rock. I'm sure it can be third class with better route finding. Some of the rock is loose, so be careful, especially if you have partners below you.
