The cheapest way from Tokyo to Sapporo is by train with the
Seishun 18 kippu.
On the page they suggest a route that is possible in 2 days, i took 3 ( as you will have to buy 5 days anyway, thatll leave you with 2 after arriving in Sapporo). I stopped in Morioka and Hakodate and stayed in Hotels, so after all its maybe not that much cheaper than flying but you spent 2-3 days.
In Sapporo you will not be able to camp because it is a big city, you can however stay in youth hostels or capsule hotels starting for 3000yen a night, there was a nice new place in downtown when i was there, i can look up the name for you if you want (it was kinda fanzy kapsulehotel).
You can probably camp in most other places around Hokkaido, on beaches or stuff, but you problem will be transportation... If you could speak a bit japanese i`d recommend hitchhiking (was quite easy in Hokkaido). You can still do it without japanese, people will pick you up! If you could get the seishun 18 kippu and make it to Sapporo in 3 days you could use 2 more days in the train for longer trips and do the rest with hitchhiking or buses.