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:
By completing everything that there is to basically offer, like completing the main game and then turning back to finish up whatever's left (like the rest of the tournaments). Just casually go through the game, doing everything you have to, and it should come naturally. As a note, when you start Challenge Mode, you'll be given a Bolt Multiplier that goes up with successful enemies slain without being hit.