We would also recommend that you fly and rent a Class C out of Alaska because of your distance and time constraints. It would be a much more relaxing trip for all of you. You will also be spending a lot of money from Florida to Alaska on fuel, food, campgrounds so apply that money to the fly/rental and you'll get to spend more time actually seeing Alaska. Definitely, for your first time, there is much to see just on your way in the lower 48, then through Canada and the Yukon but if you don't have the time to stop and explore and putting in long driving days that would be boring to us and I can't imagine the kids would enjoy the long driving days and passing up things along the way.
As mentioned, driving, the trip would cost you $10,000+ from Florida so compare flying and renting.