Puerto Rico is part of the US, so travelling from San Juan to Dallas is not considered an international flight. But there is an internal immigration checkpoint whenever travelling from Puerto Rico, Guam, the US Virgin Islands, or the CNMI to the 50 states or DC (and only in that direction). That checkpoint is at departure, so you don't need to worry about going through immigration in Dallas.
At the checkpoint, travellers who are not US citizens must establish admissibility to the US. I assume that as an Australian citizen, you will travel to the US using the Visa Waiver Program. In that case, when you depart from Puerto Rico, the fact that you have a sufficient amount of time remaining in your 90 days of VWP admission will be considered enough to establish admissibility.
This is relevant because individuals who are living in the US without a lawful status, if they choose to travel to Puerto Rico, might find themselves unable to travel from Puerto Rico to elsewhere in the US.