Tom, you could do it however you would need some sort of backup vehicle to carry stuff - there is a LOT of unihabited country on the Darwin-Adelaide road.

Best time of year would be our cooler months (June-September) if you are starting up north because otherwise you have the monsoon season and cyclones (hurricanes) to contend with.

Your biggest hazard on the road will be road trains - these are semi-trailers pulling several trailers and they move at high speeds. They also kick up a hell of a lot of dust. All the road safety advertisements for the Northern Territory advise car drivers to pull over and slow down when road trains pass and wait for the dust to settle. As you can imagine, on a bicycle you will be much more vulnerable to the effects of these mechanical behemoths.

With enough advance notice, you might even get some locals who are crazy enough to join you.