This achievement is gained once a creeper has been defeated if the killing blow was done by shooting it with arrows. This must be done in survival mode. You cannot do this on peaceful.
Easy difficult should make this achievement easier to get.
To do this you need a bow and at least one arrow.
To craft a bow you need three sticks and three string. String is a drop from spiders and spider webs:
To craft an arrow you need a flint, stick, and feather. Flint is a drop from gravel, and feather is a drop from chickens.
You can also find arrows as drops from skeletons (this is easier).
Once you have your bow and arrows, go and find a creeper then kill it. They will spawn at night time on the surface, so head out at night. Shoot the creeper until it dies.