You should be able to find Plague Hearts pretty easily around the map as you play, but you also have the option of locating them via the radio for 350 influence.
Each time you kill a Plague Heart, they get progressively stronger, with more zombies as protection. However, it's still pretty easy as long as you have distractions. Explosives do the most damage, but are often tough to aim in close quarters where you'll usually find the Plague Hearts. You'll also want to bring guns and ammo. Your melee weapon will be backup. The best strategy is to place a radio outside of the building the Plague Heart is in, sneak inside to the Plague Heart, then activate the radio remotely. If you don't have a radio, fireworks is the next best option, because they take time to start up, which will give you time to get near the plague heart. Once the noise starts, attack the plague heart. Throw bombs, shoot it, hit it with your melee weapon. Keep an eye on your mini-map to see when your noisemaker stops working, and then toss some firecrackers or set down a stationary one away from the heart, then go back and keep hitting the Plague Heart. You can see the pulsing of your noisemaker on the minimap, so even if your busy shooting and bombing the heart and can't hear the noisemaker you can see when it stops working.
If you don't have noisemakers, it gets a lot harder, but not impossible. You just have to alternate between killing zombies and hitting the plague heart. You'll need lots of meds and lots of snacks, but it can be done. If you find yourself in this situation, bring along another community member, or fire up a flare and call for some multiplayer help.
The Plague Hearts let out gas every now and then that can increase your infection meter, however you can hear and see it when it happens, so all you really need to do is get out of the way for a few seconds until it clears. In the below video, my gun jammed, and I was still able to defeat the Plague Heart with just a melee weapon. This was Plague Heart #7 in my game, so it's doable even at this level with a melee weapon.
The first time you kill a Plague Heart, you'll get this achievement:
Do this four more times and this achievement is yours.
Obviously all of this is easier with multiplayer, but as you can see below, even single player isn't very hard.