South and east will be the depths of winter: grey and rainy. Perth in winter gets rain too but often has nice clear cold sunny periods.
If you go to the tropical areas like Cairns, it will be great that time of year. Since you have so much time, I would suggest doing the north of WA: Kimberleys, Lake Argyll, Karijini Gorge, etc. The weather will be beautiful and the scenery there is quite spectacular.
The outback is a good bet for it not raining, but you can expect nights as bad as -10C with no facilities other than whatever tent/bag/vehicle you bring.
Hell, with a month, you could do a Perth->Darwin->Adelaide trek and see so much beautiful stuff.