I have driven a fair bit of Australia using caravans and campervans but not all around in one hit ( something yet to be done ) but to get best weather in May/June, I would start in Brisbane and head north, you able to get as far as Cooktown on sealed roads ( and if you are hiring, without a 4WD campervan, you will be restricted to sealed roads ). ---- change that to Darwin --- see below
So you keep heading anti-clockwise, popping down to Alice Springs and Uluru before a backtrack up to Darwin and then on around to Kununurra and Broome before heading down the west coast of WA as Spring is coming on.
Lots of wild flowers - http://www.rainbowcoast.com.au/areas/rainbowcoast/wildflowers.htm .
There are going to be lots of places where you will have restricted travel with a campervan and so in some places you might want to park it up and do a tour, that particularly for places like Kakadu in the NT, the NW of WA and Fraser Island.
Kakadu can dry out fairly quickly after a wet season and getting there even in July can find it a bit disappointing so just thinking on that score, an alternative to what I started you on would be to arrive in Darwn, explore locally to recover from jetlag and then take a tour to Kakadu before getting a campervan.
Head down to Alice Springs and Uluru before backtracking to head around into the NW of WA and you will find you are possibly even better off with wildflowers as you head south to Perth and beyond for September/October and starting the easterly trek with westerly winds pushing you.
With that timeframe, you will be arriving over east at a time when there are a lot of various festivals on so do a bit of research on that if of interest, The Melbourne Cup being on first Tuesday in November, it the 6th for 2018 and there is a whole Spring Racing Carnival spread over a few weeks.
It is also a time that flowers around all eastern cities will be blooming so a lovely time to be about generally.
Even though you can get Spring and Summer storms, you can kind of dodge those to some extent with weather forecasts if you are on the move and November to April is a great time for the south east and Tasmania.
In heading north towards Brisbane and Cairns in April to arrive up at Cairns/Cooktown a year from when you started, you will miss the cyclone season and worst of humidity.
There is not a great lot between Cairns and Darwin so if you elected to give that stretch a miss and end your journey in Cairns, you will have arguably had best travelling weather throughout the 12 months and missed what is arguably the least interesting bit to do.