There are many kinds of campaigns you can run and many ways to describe a campaign. However, people tend to have certain ideas, themes, or enemies they prefer to use. In my case, it's undead. As a Dungeon Master I like using them as my baddies and as a player I like going up against them. Undeath is a very general idea that allows for a lot of freedom. Since I've got quite a lot of experiences and opinions on the matter, I'm hoping that working through some of it here will help people who are considering running such a campaign in the future.
Some themes naturally come with undead. Life and death, to some extent right and wrong, the idea of forever, corruption, inevitability, fear, and what is natural are ones that quickly come to mind. However, we have many other options as well. One of the great things about undead is they can be used with other kinds of themes in supporting positions. Anything that can die can become undead. That gives us an incredibly large set of creatures that we can draw on and also themes to explore. A necromancer could have an undead army, but their goals could touch on many other themes not yet listed.
Undead Are Typically Evil
In role-playing games, undead are typically evil. There are exceptions in some adventures, but more often than not they are evil. Becoming a lich turns a creature evil. Becoming a vampire does the same. Playing with this little element can have interesting implications. In a world where the existence of the afterlife is known for sure, and methods to get there are known well as well, what kind of person would willingly choose to remain in the world as a vampire and lich? For what reasons? Exploring this question could lead you to campaign after campaign and villain after villain. Could people use skeleton guards for good? Good necromancers can be a lot of fun, especially when your players see necromancer and automatically think evil. Playing with those expectations without coming off as a jerk might take some skill and experience though.
When dealing with undead centred campaigns, there are a few world questions that come up quickly. How are undead created? Why would they be created? What do people think of undead? There could be a kingdom that relies on necromancy for its armies. Otherwise they'd be wiped out. Clearly they would not think negatively of necromancy. Their enemies though? I think that part is obvious. What happens if someone gets caught with undead? What if they get caught using necromancy? Are vampires known or simply the stuff of stories?
Undead, even without getting into the ability to turn any creature into a skeleton, zombie, or other version of itself, have quite a variety of different badies that can be thrown at your players. Even just looking at big bads, there are many options. Vampires, liches, death knights, wights, and revenants all provide different ways to make a campaign awesome. Take revenants. Right off the bad, you can make your party out a group of revenants out for revenge against the big bad. You could also have the opposite where your party is being chased by a revenant and they don't know why. This one can be a bit tough if players don't like you trampling on their backstory (it's very valid not to like having your backstory partially re-written as a player but some players are fine or even like it) but there are other ways to make it work, such as a helper NPC that they are traveling with (if this is the case, it's best to have 2 or 3 so it's not obvious who it is). The possibilities are really quite vast and varied depending on which one you pick. Just look at vampires and Ravenloft. Some monsters are obviously more inspiring than others, though.
Large Body of Work
There is really quite a large body of work you can draw on when dealing with undead. There are many legends surrounding them, many books that can help inspire you, and a massive amount of horror movies to draw inspiration from. Though it may seem weird to shamelessly steal from these sources, I find it does make it much easier for me to find inspiration when I need it. It could be as small as encounters or as large as campaign ideas (everyone has watched a movie and thought “I think it would be better like”) but in general, getting those creative juices going can be one of the hardest parts.