When I was growing up, we always drove from Upstate NY (~20 hours), split into two pieces with an overnight (or just a pull-over-at-the-rest-stop-and-sleep-in-the-van) break built in. This was in the days before portable DVD players or the Nintendo DS. Walkmans existed, but we weren't allowed to bring them. It was just Mom, Dad, me, my sister, an old 8-track player, a CB radio, and whatever books and games my sister and I brought to amuse ourselves with.
Honestly, I have as many great memories from the road trips as I do from the days at the parks. We sang, laughed, shared memories, and generally had some great family bonding. I'm sure it helped that our vehicles were always relatively cushy conversion vans with comfortable reclining captains' chairs for each person, and even a little table in the back that my sister and I could set between the chairs to play games. Also, my Dad loves to drive. My Mom was a Type I diabetic, so we had to stop at a lot of restrooms, but it added to the adventure. In fact, at one point when I was about 8, I actually began keeping a journal of every restroom we visited on the way down (size, cleanliness, availability of TP, etc.), with the idea that it would be an indispensible guide the next time we went, LOL!
At least it helped keep me occupied during all those hours in the car.
As far as the "schedule," we used to leave on Saturday morning, drive all day (stopping for dinner), stop for a few hours to sleep, and then Dad would wake up super early to resume driving while the rest of us slept (around 5am or so) and we'd be coasting into the Fort Wilderness Campground before lunchtime on Sunday.
If you're up for the challenge, it's totally doable -- and will likely save you hundreds of dollars, if you're into that sort of thing!