This is a cumulative achievement. As long as you break as many crates, destroy enough enemies, and run over any weeds poking out of the ground, you shouldn't have any trouble getting this early on. There are other versions of this trophy:
As a grasp, I was just going through the process of clearing everything when it became available. By the time I entered the Agorian Battleship, clearing the rounds in the first tournament, I had this trophy, so it would be your best bet to clear everything and take it as it comes.