Just a couple of additional comments. The actual costs for us, tends to be about $7,000, if we knock off about $3,000 that we would spend even if we stayed home, for food, utilities, etc.
You can calculate your fixed costs and then look at your variable costs. From here in south Florida, the distance to Fairbanks is roughly 4,500 miles or 9,000 for the round trip if we went the way a bird flies. But we don't, as we swing over here and there to see places or people, run round in Alaska, etc.
An average trip, if there is such a thing, I budget for a trip of between 13,000 and 14,000 miles. Some trips are less, some are more, depending on our route. Fuel is the largest fixed expense we have and it is not hard to figure the approximate cost of that. With my previous truck, a Dodge Cummins with 235HP, it would be close to 1,000 gallons of diesel for the trip. Now with my Chevy truck, larger, heavier and more power (397HP) I will have to budget for about 1,100 to 1,200 gallons of diesel. So with an average of $5usd per gallon, my fuel costs will be between $5,500 and $6,000. Not much I can do about those costs, other than to make the trip as straight line as possible, etc.
Fuel costs are the same for a traveler regardless of the number of days it takes to get to your destination. If a person takes 10 days of driving to get to Alaska, like I tend to do, the camping costs are reduced but not the fuel costs. Boondocking is probably the best way to save some on the trip, and it gets much easier to find impromptu places to spend the night as you get farther north, beside a lake, stream, a roadside pull off, etc. Our food costs don't vary a great deal whether we are traveling or staying home. If we had to, we could reduce the food cost for both, I am sure.
There is no getting around the fact that a trip to Alaska or any trip of an equal distance, is going to be expensive. Most of us drive self contained RVs, so a mix of campgrounds and pull off spots works for most. The weather is also a factor, if it is a hot summer, you can anticipate many forest fires and resulting smoke, making running the AC important at times. (for us anyways) On the average in a 7 night week, we will stay about 3 nights in a campground and the other 4 boon docking for free. We may spend a week at a commercial campground in Fairbanks, then spend a week for free camping in the Tangle Lakes area west of Paxson Alaska.
Before my wife and I got married in 1973, I had made 4 or 5 car trips to/from Alaska and Oklahoma where my parents lived. Several of those trips I had to stop and find work to be able to afford to continue the trip. The Canadian authorities at that time required you to have $300 with you which you had to show, at least I did. My camping gear consisted of an Army surplus tent, sleeping bag and a tarp, a plate and a skillet and coffee pot and cup. Fuel and some food were my only expenses. A couple of those trips I made and never spent a dime for lodging the entire trip. Sleeping in the car, sleeping beside the car wrapped up in the tarp, or setting up the tent beside the car. Great memories and I sure had fun, met a lot of interesting people. In the 1960s, many of the abandoned construction buildings used by the Alaska Hwy builders were still standing near the road, so these became impromptu "motels" for many travelers. A different age, a safer age, etc. Those old building have all disappeared now days.
If, I was traveling with a car that got 30 mpg, fuel costs would drop to about $1,500 for the trip if I could hold the mileage to about 10,000 miles. Sleep in a tent or tarp again, eat frugally and I would guess I could do the trip for under $2,500 but my wife wouldn't go with me. LOL Her idea of roughing it is slow room service in a 5 star hotel. That is what happens when you marry a city girl I guess.