This can be done most easily with Australia as there is only one way into the continent. I started a game with 2 AI players and random territory distribution. You're looking for at LEAST one, preferably two or more starting territories in Australia, if you don't get them you can just restart fairly quickly and try again. Hopefully you'll play first so that you can dump all your troops into your Australian territories, and go on to conquer the rest. Once you have the continent, move as many troops as possible to the territory leading into Australia to use as a blocker.
Don't recall if the trophy pops as soon as you take over the continent on your first turn or after the turn has ended and you still have control of it. I got this in the first game I ever played, it shouldn't take long to replicate with the method above.