You can't kill a zombie or go down, but only after you entered the first zipline and you're planning on doing them all. You can do everything just fine until you're ready. That includes killing, going down and riding ziplines. So don't get confused by the trophy description.
In order to go up the ziplines you need the zipline handle. The Hermit will give it to you, but only after you've grabbed and turned the two zipline cranks around the map. There's one near the front of the boat, near the Soda perk, and one at the very back of the other boat.
Check this short video for the spawns:
There's also a crank spawn inside the little room at the Stern.
Place them near some ziplines ( Begin of #5 and end of #8) and crank them until the ziplines are correctly aligned. Then the hermit will give you the zipline handle, allowing you to go to the facility.
There are 9 ziplines in total, in this video I show all of them in a row (and me unlocking the trophy):