I day-hiked it last January, but that wasn’t a normal snow year. I was able to drive fairly close to the portal. It took 12:40 from where I parked to the summit and back. There is a trip report at
http://www.whitneyzone.com/wz/ubbthreads.php/topics/34913#Post34913.
You don’t mention whether you have climbed the Mountaineers’ Route. If you haven’t, you should do it at least once before you try it in winter. A summer trip and another trip with a moderate amount of snow (November, or May?) would be good preparation.
Every winter, we get a few long periods of nice weather. Going towards the end of one of these periods can make a huge difference. A warm, sunny day after 10 days of nice weather to consolidate the snow really helps. You don’t want to be there during or right after a big storm that leaves several feet of avalanche-prone snow. Pick your dates carefully, and reschedule if conditions aren’t right.
For many years, the County has placed a “ROAD CLOSED” sign across the road during snow season. They have usually looked the other way when people drove around the sign to access Mt. Whitney. However, quite a few idiots have gotten stuck in the last few years, blocking the road for weeks or months. Last spring, they installed a gate across the road, making it impossible to drive the last few miles to Whitney Portal. I heard that climbers had asked the County to reconsider the locked gate, but I don’t know what came of that. You will want to know the status of the gate before you go.
Good luck.