Camping is still one of the cheapest forms of accommodation in Tasmania (or anywhere) and there are so many amazing places to pitch a tent. You can find camping spots everywhere from council designated camping grounds to many of Tasmania's national parks. Go See Australia has a handy directory of caravan and camping sites around Tasmania. Check out the parks website for information about camping in national parks.
Make sure you have the right gear for your camping trip, exposure can be a real danger so having a tent that can hold up to strong winds is an absolute must. A sleeping bag capable of keeping you warm in sub freezing temperatures is also strongly recommended as exposure can be a killer in Tasmania. Do not underestimate the weather, even though it's summer the weather especially in more mountainous areas can turn quickly from a beautiful day to dumping snow on you. Always make sure you have cold weather gear and wet weather gear. When camping near rivers make sure you are well back and higher up from the bank, river levels can rise quickly so don't put yourself in a situation where you can get washed away. Make sure you aren't camping on private property, it's a great way to annoy the crap out of the locals and do yourself no favours, plus farmers, livestock, motorbikes and (possibly even) shotguns aren't necessarily a good combination. If you aren't sure ask a local where the best places to camp are.
If you are looking for private camping spots have a look at Home Camp, the site is basically AirBnB but for places you can pitch a tent.