I did this on the first big boss Lady White. The time was very tight so you need to be on offense as much as you can. The pig is your best player for his jump attack but you want to have him for the 2nd and 3rd phase so I used the monkey because of his range.
Phase 1. If it’s only a couple vines that pop up, attack her from the side of the vines so you can still hit her because of the monkey’s range. If there’s a bunch of vines you’ll need to stand in the middle and wait and time your double jump and dash up at the right time so you can hit her but not get his yourslef. Biggest problem is the cannons that shoot from the side because they can kill your character quickly so I would change to the monk or the green guy during this attack. Try and dodge what you can but you have to hit her. If you’re not losing a guy during this phase I might say you’re not aggressive enough.
Phase 2. Her attacks are very simple to dodge and run away from. Once she’s about ready to pull up the vines switch to the pig, once the vines are fully up (don’t hit too early or it wont count) hit the left and right branch. Wall jump to the top, jump and spam the X button and land on a branch. Not only does the pig do a lot of damage but his range should destroy the floating balls that come after you. Jump and spam the X button again, do this a third time but after you’ve finished your move set dash up or to the place you think she’ll move. Hopefully she’s close and you can hit her again by spamming X. You’ll have to do this whole process one more time. This has to be done with her pulling up the vines just twice or you won’t make the time to beat.
Phase 3. The boss’s attacks are pretty much the same but I still switched to the monk or green guy in case I got hit. Once she’s about ready to pull a vine out from each side of the screen I switched to the pig. Hit the first branch and jump on it. Hit the the branch above it and jump on it. Just like in phase 2 you’ll want the spam the X button and go through your move set 3 times and on the third time after your attacks are done dash and try and get some more hits in. You’ll likely be hit on your 4th set of attacks but that’s ok bc your pig shouldn’t have been hit yet. Do this a second time and if you’re very lucky you can defeat her. I kept getting hit on my fourth set and needed a third time to jump up on vines to attack her. My time was 4:53. I tried the monkeys special attack in the third phase once but it didn’t do enough damage.
*Edit- the developers just posted the emerald times to beat on every level so based on those you can also get this achievement on The Doomed Jungle, The Iron Fan and King Bull and some others.