To be eligible for teaching work in Australia, you’ll need to go through the full registration process for the state you plan to work in. You may need to get your education and training qualifications officially recognised in Australia first, and that can sometimes mean a long wait. It is best to see to that right away. Once you are a registered teacher in one of the Australian States or Territories, you can apply for jobs right away. Many people find it difficult to get interviews when they are still based overseas, but try to arrange as many as possible to minimise travel costs. There are several good recruitment agencies for education and teaching careers in Australia, such as TeachAnywhere or Smart Teachers. Google a few different agencies, send them your CV and see what they recommend.Some state school departments advertise urgent vacancies, so check there first. It never hurts if the employer is a little desperate. Don’t forget to look for jobs at independent and private schools, or the various government education departments.
Also, search online forums for teaching in Australia threads. You might try having a look at the Teaching Overseas Sub-forum on TES Forums, or try BritishExpats.com/forums, Seek.com, Career One, or Teachers.on.net.