In order to get this achievement, you need to both
find and
destroy a plague heart
in the same night.
This is far easier to do with one of the first three or so hearts, because as you destroy a plague heart, the next one gets stronger, and that strength stacks, so by the sixth or seventh, it can get tough to pop this achievement.
Fill up on ammo, distractions for the waves of zombies you'll have to deal with, meds, and explosives, leave at dark, get in a vehicle, and start driving. Scout out locations, and as soon as you find a plague heart location, attack! You'll need to destroy the heart you find before morning. Toss some firecrackers or whatever distraction device you have on you to distract the mob of plague zombies, and attack that heart. Another option is to set down a radio, sneak in to the plague heart, then activate the radio. Fireworks also work well because they take time to start up, giving you time to sneak in to the heart. If you keep an eye on the minimap, you'll also see the pulsating of the noisemaker, so you can see visually when it stops working, to prompt you to toss or set up another noisemaker.
The first three or so plague hearts are very easy. The first one can really be destroyed easily with a melee weapon (the heart will release poison gas, but the first two hearts still shouldn't be a problem as long as you have meds), but a gun or explosives will make it even quicker and easier, and avoids the gas. Regardless, the first three or so shouldn't be much of a challenge. Just kill the heart before morning, and you'll get the achievement.
Sometimes destroying plague hearts comes as a task to help other communities, so if you locate the plague heart at night via the mission, you can also get this achievement if you finish before the morning.
And if it's the first plague heart, you'll also get:
This video shows one strategy to defeat a Plague Heart. I did it during the day to make it easier to see what's going on. This was Plague Heart #7, so you can see that even at this level, it isn't that hard to do. Set up a distraction, attack, repeat: