Complete a "Passport Application Form". These are on the website of the Australian Passport authority If you are overseas when applying for the new passport, complete an "Overseas Passport Application Form."
Provide the birth certificate of the child, as well as his previous Australian passport. If the child was not born in Australia, provide the birth certificate, with an English translation attached and proof of Australian citizenship. If the child does not have a birth certificate, then you must complete a form "B6."
Provide details of the person completing the application. This will usually be the parent or guardian of the child. Consent will also be needed from both parents, which can be given separately in any Australian embassy or consulate. If you cannot obtain this then include all court details referring to custody and responsibility of the child, as well as a form B8 or B9, with the application.
Send the form, with all the appropriate details, including photographs, to an Australian Passport Office or an Australian diplomatic mission overseas. Once you have done this you can track your application online and make payments when appropriate. It will take around ten working days for the Australian Passport agency to process a child's passport, assuming all the documentation is correct.