Hi Maddy! The Australian Constitution gives the Governor-General in Council the power to appoint High Court of Australia judges. This means the Governor-General appoints High Court judges on the advice of the Australian Government.
In practice, the Prime Minister, on the recommendation of the Attorney-General (the minister responsible for legal issues) and the Cabinet, chooses the person to be appointed. The Attorney-General consults with the Attorneys-General of state governments, and with senior judges and lawyers before a decision is made.
There are no qualifications for High Court judges other than that they must be under the compulsory retirement age of 70 (s72). It is usual for High Court judges to have had long and distinguished legal careers before they are appointed.