This sounds like a hack & slay adventure.
Why are the undeads attacking? What does the leader want?
Why does the Grey King rise this days and not earlier?
What happens when the PCs just head elsewere and ignore the quest?
Give them an interest to fight the undeads.... maybe they surprised the PCs on resting and one of them got infected, slowly turning to undead. Do they want a cure for their companion?
Build in some twists in your story!
My group only meets once every 2-3 weeks and is comprised of very casual gamers, so I find that combat and exploration works better than in-depth social, political, interpersonal stuff, although I like to build up interesting back-stories and lore for them to figure out.
But yeah, those are good questions and ones I am asking and have vague ideas for, I just didn't want to go into too much detail in that post. But to add a bit, the Grey King is the undead ruler of the Tainted Wood, which was once called the Golden Wood and an elven nation that was allied to the human kingdom of the region a few hundred years ago. The human king become power-hungry and obsessed with his expanding his borders, and started pushing his allies around - including the elves. A long war commenced, and the end result was the Tainting of the Golden Wood, which became dark and treacherous. All elves left, although there have been rumors of dark and shadowy elves that still live their and kill all people who enter. Anyhow, in recent years a wizard--a necromancer, perhaps--entered the Tainted Wood and found the tomb of the former ruler of the Golden Wood. He raised him from the dead, and the Grey King began gathering his host, desiring revenge on humankind...
That sort of thing. I'm imagining that the "walking dead" are the result of the necromancer's interest in gaining power, and his unholy alliance with the Grey King.
The interest in fighting the undead would initially be to defend their home region. This is built-in incentive to not head elsewhere, but if they do I'm fine with that - I'm hoping for a sandboxy feel to the campaign, but of course the necromancer and Grey King will still have their plans, and perhaps if the PCs return to their home they'll find it over-run by undead.