As you make your way through Greenpath, you'll reach the following circled area:
Here you will find Zote being held in the jaws of a Vengefly King:
For this achievement, simply ignore Zote, essentially leaving him to die. Return to this spot later and you will find Zote's skull on the platform. Attack it and the achievement will unlock.
I don't know the exact trigger but I went back to check after finding the Crystal Heart ability in Crystal Peak and Zote was dead.
This achievement is in conflict with the Rivalry achievement, which you can only get if you save Zote when you first see him: