Each of the levels has a section broken up into multiple sections or chapters. The achievement is to play through 4 different sections, and they don't have to be back-to-back. If you're planning on playing through the campaign, this will likely come pretty naturally, as there are some sections that are easier or where your character is less in direct danger and needing to heal.
That said, if you're hell-bent on grinding this out as soon as possible, be sure to use/abuse the "Takedown" ability. This charges up as you kill zombies, typically about every 20 enemies or so, and will be ready when the section in the HUD lights up. You'll also only see the message "Press 'A' to execute takedown" if the takedown is charged and ready.
Executing the takedown means you'll recover about 20% health as well as a small amount of ammo. Keeping your health topped off in this way means you're far less likely to require a medkit.
Additionally, starting either late in Level 2 or early in Level 3, you'll start discovering Divine Grenades. These will create a (gooey) area of effect that can heal you to nearly full health, so these are a cheeky way to recover health without resorting to medkits as well.