You unlock supers in the mausoleums on each island. Mausoleum 1 is accessible pretty early in the game: if you beat the first run-and-gun level you see ("Forest Follies"), a bridge will open to your right, and if you follow the path past the bridge down and through some trees you'll see the mausoleum next to a fishing fish.
Beating the mausoleum is pretty straightforward - just parry every ghost - and you'll have your first super. Equip it (bring up your equip card by pressing Y while wandering the hub world), and use it to beat a boss. You trigger your super with your "Ex Shoot" button (B by default), when you have all five cards collected at the bottom left of your screen. These cards build up slowly as you shoot things, or quickly as you parry pink objects.
The easiest boss is the one in "Botanic Panic". The only real issue here is that the killing blow has to come from your super, and you'll be guessing whether the boss is about to die or not since they have no health bars.
Your mileage may vary, but here's what I did:
*** Spoiler - click to reveal ***Boss 1/3 shoots horizontal attacks at you, just keep facing him and jump over each one individually; the fourth of each group is parryable. Use those to build your super.
It's no big deal if you miss one.
Boss 2/3 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 your super by the end of this boss, almost certainly.
Boss 3/3 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 delivered my super, which killed him immediately. The achievement unlocked right when the screen said "KNOCKOUT!"
EDIT: Reportedly this can be earned on "Simple" difficulty, which was confirmed by TA user HawkeyeBarry20 in the comments. Thanks!
Good luck!