Your "Extra Special" move is the one mapped to your "Ex Shoot" attack (B by default). This changes to your super attack if you have all five cards on your meter (bottom left of screen, gained by shooting enemies and parrying). As a result, you really want to do this before you've earned and equipped your super (gained at the mausoleum).
After you have a super, it doesn't look like you can unequip it. That means that any time you gain all five cards, instead of being able to spam your extra special move, you shoot off your super attack, which won't get you this achievement and drains all your cards.If you've already earned your super, you'll want to unequip it so that you don't fire your super and drain your entire meter when it's full. Unequip it by selecting it and pressing "Y" in the power selection screen. [This hot tip brought to you by TA user ACBL80, who can read better than I can.]
So this is best done right away, at the very beginning of the game. If you do have your super, make sure you routinely fire off your special to use a card and keep the meter below 5 full cards. Otherwise, it's fairly straightforward using the first boss battle ("Botanic Panic," against the Root Pack). It can be done on "Simple" difficulty.
*** Spoiler - click to reveal ***Boss 1 of 3 shoots horizontal attacks at you, just keep facing him and jump over each one individually; the fourth of each group is parryable. You can use those to build your super, but it's not important if you get them or not.
optional: Boss 2 of 3 triggers raindrop attacks that are slow and easy to dodge but hard to parry. You don't really need to parry any of them, though. Just keep shooting. You'll have maxed out your super meter by the end of this boss, almost certainly, unless you've been firing it off every once in a while. This boss doesn't appear on "Simple".
Boss 3 of 3 calls down homing carrots and occasionally attacks with a third eye that shoots light at you. Shoot the carrots quickly rather than waiting until there are a lot, keep on the move when he's shooting out of his eye, and otherwise focus fire on him.
I waited until right after his third eye attack and then spammed all four of my specials, which killed him immediately. The achievement unlocked right when the screen said "KNOCKOUT!"
EDIT: TA user JORAX mentions that he had trouble getting the achievement when using the special move for the spread shot, but it popped without issue using the default weapon, so try that if you have issues.