A week is definitely not much for a country-continent like Australia, but after a quick check I saw that the drive from Sydney to Brisbane is just over a thousand kilometres in a straight line. Perfectly doable solo in seven days, with enough room to make stops along the way. Of course, I’d recommend taking more time if you can (longer road trips here are always worth it) but if you happen to have a week, this route is a great option.
I rented a car with Sunny Cars, fully insured (always a comfort when you’re covering long distances on the other side of the world). The roads were straightforward and easy to drive, even solo. Most of the time it’s just long, straight stretches with the occasional turn-off if you want to stop somewhere.