You firast start by writing down what and who you know about your family as far back as you can remember, then ask your parents and older relatives to add their information, ask your eldest living relative if they have any certificates or papers (e.g. copy wills) from the earlier generations. Once you have reached back to around 1901 you can start using the online census indexes to locate whole families and with the Civil registration index on line you will eventually get back to the irish ancestors that were supposedly deported.
What i would say is that it is possible the family had assisted parish to Australia by their parish council, this was common during the very severe poverty years which occurred twice in the 1800s, when the landowners refused the poor the access to produce in prefernece to selling the crops and then the 'famine' worsened dur to potatoe blight ruining the local crops.