The Great Traverse by car?

Good evening
My name is Leonardo and i'm living in Australia at the moment
I'm planning a trip around the Japan for the next March/April and i want to do the Great Traverse
My question is; is It cheaper move from city-to-city with the JR or by car? Because my trip probably will take 60-90days and if i buy a camper i can put a bed inside It and sleep there, without booking hotels
Any info will be appreciate,thanks! :)

Pretty open ended question. So only generalisations can be make for an answer. If you are taking 60-90 days, then you'll have a fairly slow pace, and adding in many more sights along the way. So I would be thinking of multiple forms of transport. Not just option a vs option b.

Car toll fees will rack up very quickly considering the distance you wanna cover. And JR can get expensive depending on how fast you wanna travel. But without and itinerary or rough plan, that isn't something that can be answered concisely.

I think buses would probably be a good option for your travels. AS they are the cheapest form of long distance travel, but sometimes the slowest.

I'd start with a plan of sights you wanna see and a vague itinerary idea, then begin to plan the transport options.

How much are you going to spend on the camper?

If you buy it, what are you going to do with it afterwards?

If you are going to re-sell it, how long will that take, and will you have to stay in Japan during that time?

How much are you going to be spending in gas for the duration of your trip?

Four 21-day JR passes and one 7-day pass cover 91 days for $3332 AUD. Something to consider.

There are different JR passes available- meaning you are not confined to the standard JR pass (7, 14, 21 days) that requires you moving every day.
For example the JR east pass allows 5 or 6 train rides within 14 days and the JR East South also offers the same flexibility. Regional passes cover specific parts of the country.
Hence as others have suggested, decide which part of the country you want to travel to and look at what the various passes can offer in addition to the long distance buses. I have no experience on long distance buses but they look clean and comfortable. Also I have not read of accidents like in some other nearby countries.
Cars and campers may save you money in terms of accomodation but in reality parking charges, tolls and petrol may negate all the perceived savings.
Hostels, business hotels are options for accommodation. It would be difficult Ifd imagine to park a camper van in the city or sleep in it.
A combination of JR passes, buses and purchasing separate train tickets could probably be a good outcome.
Use hyperdia.com to calculate the fares.

I suspect the OP thinks there are camping grounds around, and while it might be a good way to travel in Australia and some other countries, not so practical in Japan. Parking in parking building or a small parking lot down-town doesn't seem like the greatest way to spend the night just to save a couple of hundred dollars (A$100-A$200/night is quite achievable).
Also related to the plan is not just buying and selling again, but all the messy insurance requirements.

Just on the push to use rail passes, it looks like a trip of this kind wouldn't be a good fit. The travel pace is slower, and the distances each day are correspondingly quite short. Passes pay off when you are doing lots of travel in a short time. Even on some of my longer trips/tours we still get good value out of the passes. For example, 5000km in a month a pass makes sense. 5000km over three months, no.

a non-resident can't register a car. you can buy a car, but it is just a box. you can't drive it (it is illegal).