For this achievement you'll need to take an Elder Hen/Roostro and throw them into an incinerator. There's a couple things you'll need to do before a chicken will turn into an elder.
1. Create a coop. Load it with a few Hen Hens and at least one Roostro.
2. Let it sit for a while. After a chicken lays (or for a Roostro, sires) a certain amount of eggs (chickadoos), it will become an elder version of itself.
Hen Hen: Needs to have 6 chickadoos.
Roostro: Needs to sire 18 chickadoos.
3. That's it. Once one turns into an elder, suck it into your vac pack and launch it into an incinerator (Costs 450 newbucks).
Here's a video for all achievements if you'd prefer. Skip to 16:05 for Carousel.