It all depends on what you want to see and how long you will stay. You have to think about the climate and the seasons. There are a few other things you should care about as well...

The North, like Darwin and Cairns, are very humid all through the year but especially in summers (December, January, February). The more South you go, the less humid it gets. Brisbane can be hot as well but as you're getting to Melbourne you got a quite cool climate. Like London or Amsterdam for instance.
The North is also very wet in rainseason (January, February March), so you'd better not go there that time.

The prices are also different in High Season compared with the normal rates. After finishing college and summerholiday in Europe and America (Juli, August, September, October) a lot of people are coming to Australia. Hostels charge more money than in other months. But that's not the only thing. If you want to buy a car at the carmarket, it's better to buy it in any other month. It's like we had ourselves: we wanted to sell our car in April. There wasn't really a demand so there was not really a big market, so less money, but cheaper for the ones who bought our car.

If you want to work than that's another thing you should think about. The main job that backpackers do is fruitpicking. The industry can not actually cope without them I think. You can go fruitpicking in a quite a few parts of Australia. But it all depends on the seasons again so make sure you are prepared about that.

There are of course a lot of other things that can influence the time you are going, but that would be more specific. Like whalewatching on Fraser Island; Well it has to be the season otherwise you won't see one at all...