Typically they start plowing Tioga pass road in mid-April and finish in mid-May, which leaves a short window of time that you can bike up Tioga pass road when there is little or no traffic. You can go the Yosemite National Park service website and they will usually have an update of the progress.
Although not critical, it is almost mandatory that the Whoa Nellie Deli be open at the Tioga Mart gas station. The place is amazing and you have to see it to believe it. In 2009 I was there mid-May a week before the pass opened. I rode up to just past the entrance station into Yosemite, then back down to the deli where I enjoyed their fish tacos out on the lawn in the sun. A very nice day.
You can fly right into Lee Vining, about 1/4 mile from where Tioga Pass road hits 395 and about a mile from the deli. It is by all definition a high altitude airport at 6,800 feet so be forewarned. Also, the winds can really rip in the afternoons because that airport sits right were Tioga pass opens up onto the high desert floor.