Howdy, I'm one of the two VFTTrs that Matt mentions. First, it is the Blackhead Range not the Black Dome range, even though Black Dome is the highest of the three peaks.
The most popular and common route is from Big Hollow Road (aka CR56), as far as the roads, I think the storm is supposed to go by tonight into tomorrow morning so by Friday, I think you folks would be fine for most of the roads. To get to Big Hollow, you can either get off at Exit 20 and go up Rt 32 to 32A to Palenville and then to Rt 23A west to Rt 23C in Tannersville to CR40 (maplecrest Road). That road will take you past Barnum and shortly after to Big Hollow Road. Right by Barnum is a nice hill that you should be careful on, the turn for Barnum is on this switchback. The other way is to go to exit 21 on the NYS Thruway to Rt 23 towards Cairo, go through the DEC land by the end of the escarpment trail and take some roads south to Big Hollow. That is probably the easier way to get to Big Hollow if conditions are bad. Rt 23 being a state highway, they're for sure to be plowed and you wont have to go over the hill at Barnum on Rt 40 if you're going to Big Hollow.
Enuf with the driving, shouldn't be a problem going over the hill actually, but just a warning.
As far as trails go, the climb to Thomas Cole from Barnum is fairly gradual, if you find out report "A Day on the Blackhead Range", you'll see that Dr Wu and I did a daytrip out to BH and back from about 7:15am to 5pm with plenty of stops to enjoy the views, and time to switch from crampons to snowshoes.
The climb to BH or BD from the end of BH road will be steeper and shorter. Seems like you're planning on camping? I presume at the Batavia Kill leanto? You would certainly be able to hike the three peaks on the Saturday after staying at the leanto Friday night.. Then staying there on Saturday night and taking a short hike out on Sunday? If you have any plans on sunday, you could possibly do a loop to Acra Point/Burnt knob and then take the trail back down to Big Hollow.. Would leave you enough time to get back to DC.
Jay