I think I'd get a 5-gallon jerry can. 150 miles isn't very far in some parts of the west, especially if you want to pass thru a town without filling up now and then. I do think you could make it without the extra fuel, but your mind would be on that fuel gauge half the time.
You can't "boondock" in Yellowstone -- must be in a campground.
Be sure to have proper tie-downs on your truck. For years I used the bed stake holes, but on long trips they'd often wobble just enough to loosen the chains a bit. Once, while driving in strong and gusty wind conditions on a thousand-mile trip, both of the driver's side tie-down chains got so loose that they detached. I came over a hill on I-35 in Iowa and glanced in my driver's side mirror to see the camper lifted up about two feet and still going! I slammed on the brakes and made a fairly hard right turn onto the shoulder and it settled back down. (Whew!) I drove the rest of my trip at about 30 mph and purchased a set of good, frame-mounted tie downs.
I could go on and on
...like the time I drove the pickup to my insurance agent's office to turn in a claim (from backing my truck camper into my car) and smacked the overhang/sign on the front of his office. (One of my more embarrassing moments!)
Or the time the refrigerator door came open enroute and a jar of pickles fell out and crashed onto the floor. (Pickle juice is STICKY!)
But I won't. ;)
I saw one of those a few years ago owned by the owner of body shop. I like it and have suggested here that Ford, GM and Dodge could offer such beds on new trucks. Trucks seem to get taller every year, and 5th wheel manufacturers can't seem to keep up with them.
I can't for the life of me understand why everyone is recommending I-80, ESPECIALLY when you've stated that you'd like to visit the Black Hills.
I would recommend I-90, probably after crossing Iowa on I-80/680, then taking I-29 north to Sioux Falls, SD/I-90. Traffic, especially truck traffic, will be much lighter on I-90, and it will take you directly to the Black Hills (still an excellent vacation spot). I'd also recommend a drive-through tour of Badlands National Park, 60 miles east of Rapid City. Driving through (possibly with a short hike) won't slow you much -- maybe figure an extra 2-3 hours including a stop at Wall Drug, the granddaddy of all tourist traps. ;)
The Black Hills has plenty to entertain for a week, but I'd think you'd want to pare that down to 2-3 days to leave more time for Yellowstone/Grand Teton NPs. Rather than recommend what you should see there, I'd suggest that you Google "Black Hills" and decide as a family what you want to see. Certainly Mount Rushmore, Crazy Horse Monument and Custer State Park should not be missed, but whether you want to include one or both of the NP caves (Jewel and Wind), Deadwood, Devils Tower, etc -- that depends on time available and interest.
Continue on I-90 west to Buffalo, WY, then take U.S. Hwy 16 across the Bighorn Mountains. It's a good, all season highway with not too many steep hills nor sharp curves. Continue on to Cody where again, you can choose from rodeos, the Buffalo Bill Historical Center, etc.
Sorry, I've got to go, but that'll get you started -- and nearly to Yellowstone.
I'd also recommend making the reservations at Madison Campground. It's about as well centered as any. In fact, if you need hookups I'd point to West Yellowstone (or further), which is another ~20 miles west of Madison.
As others have said, start out early in the morning. I used to visit several times each year, and we'd be out of camp before sunrise, then back in camp at 10 or 11 a.m. until 3-4 p.m., then back out until dark. We were concentrating on photography, and the best photos usually are found in early morning and late afternoon. Just so happens that's also the best time to avoid the mid-day crowds/traffic. :)