Northern sky as seen from Darwin on the morning of August 13 at 4:00 am. Click to embiggen.
The Perseid Meteor Shower peaks on the morning of Thursday August 13. Despite this being a quite reasonable meteor shower, and probably in outburst this year, for most of Australia, the radiant is below the horizon, and only the very occasional meteor shooting up from the northern horizon will be seen.
Anyone south of Brisbane will see only the occasional meteor, say maybe one or two per hour, the further north of Brisbane you are, the more meteors you will see.
You can check predictions for your local area at the NASA meteor flux estimator (choose 7 Perseids and 12-13 August 2010). People around Alice Springs and Darwin have the best chance of seeing meteors, possibly as many as one every 3 minutes. With the Moon in the evening sky, this should be a good time to see the Perseids.
To see the meteors, you will need to be up around 3:00 am local time on the 13th (yes, a really horrible hour of the morning), with best views 4:00 am-5:30 am. The meteor shower will be located due North, with the radiant just above the northern horizon. You can follow the Perseids at the International Meteor Organisations live website. Cloud cover predictions can be found at SkippySky.
A nice video of a Perseid fireball from this year is here and another one here. And here are some images from Singapore.
Note, those of you who have Stellarium, the meteor shower you see when you turn on the Meteor button in the star and planet visibility dialogue is the Leonids, the radiant is only correct for 18 November, it only coincidentally is vaguely in the area of the Perseids.
UPDATE: Meteor counts, images and videos can be found here.