Having a criminal record may impact on your ability to hold a public office, as set out below:
In Victoria a person is disqualified from becoming or remaining as a Councillor if they have been convicted of an offence committed when they were 18 or more years of age which is punishable upon first conviction for a term of imprisonment of 5 years or more anywhere in Australia. The disqualification lasts for seven years. (Section 28, Local Government Act 1989 Vic)
Under the Victorian Constitution, any person who, when over 18 has been convicted of an indictable offence (a serious crime) which is punishable by more than 5 years imprisonment may be disqualified from holding a seat as a member of State Parliament.
Most convictions arising out of nonviolent actions are not indictable, but you should check with a lawyer if you face a conviction which might exclude you.
The Commonwealth Constitution states that persons convicted of offences which are punishable by one year of imprisonment or longer cannot hold a seat in Federal Parliament. This refers to the maximum sentence, not the sentence you may have received - so if you were convicted of a crime that has a maximum sentence of 2 years imprisonment, but were just ordered to pay a fine, you may still be ineligible.