Well the short answer is you can't. According to the Australian Stock Exchange website:
"All shares listed on ASX can only be bought or sold through a broker. A stockbroker acts as your agent to buy or sell shares on your behalf, for which a fee is charged."
This statement is clear and unambiguous. However, it got me thinking. There are actually a few exceptions to this rule. The exceptions are all share purchase transactions in which the ASX is not directly involved.