Charisma for Undead? How can THAT work?To the Living, Undead do not have any charisma, they are simply d**n ugly. (Please ignore vampires and similar, we talk here mostly about lower Undead.) But we have a society composed of free-willed Undead only. Social interaction will happen, and some are heard and seen more than others.
To re-work Charisma for Undead, we must analyze their society and social-being, and to do it, we must delve a bit in their emotional baggage. Please note that Undead are somewhat emotionally-impaired. Were they not, they would probably all be dead now, or mad from this world, from commiting suicides to blind killing-sprees.
Negative emotions?Undead, even those with free will, are not friendly creatures. The transport to Undead World was but a traumatic event. Most of the Undead were torn from their master and commander. All of them were pushed forward with a burning hatred to the Living. There are no Living nearby, the hate grows weaker, and Undead begin to remember their former lives, and their masters.
Pain and Fear: Undead do not truly feel pain. There are exceptions, like holy water and the Sun. With no holy water around, Sun is the only thing Undead fear. On the other hand, it is the fastest way to die. (And some do commit in this way a suicide.) Pain and fear can but come from their background, when they were still Living. In Undead World, many old memories awaken again. See also
Self Preservation as a possible source of fear.
Hate: No Living to hate. As before, sources of hate can come from the background, too.
Positive emotions?Not many.

If you rediscover parts of your past, you may remember things that used to make you happy. Some of them may be possible to repeat in Undead World, at least partly. Thus if you find someone doing some nonsensical activity, it may be trying to find the joy from the living days again. Many Undead try to return to jobs they had in Life. Some succeed, few are truly happy, or at least content.
(Reminds me of a particular zombie, that used to be a trapper, most probably. These days you can discover his work, as you fall into a hole and your foot gets stuck. This is a BIT dangerous, of course, if you don't manage to get free till sunrise.)
Self-preservationCommon Undead are weakly equipped in this regard, but Undead World changes much. The freedom gained here, and returning memories make them feel a bit different, almost like having value, and in few cases having found The Meaning Of Life.
It should be noted that combat is deadly, and chances to recover are few and limited. Any conflict, even with much weaker foes can damage you. While the wound may be not big, it may be beyond repair or healing.
Why do anything, anyway?Once there is some shelter from the Sun, Undead don't need anything tu survive. (survive

) Some of them, mostly the Mindless Ones, do exactly that: nothing. It is easy to stay in your hole, maybe remove the dust swept in from time to time. Some fresh or old philosophers live similarly, having finally found the time to ponder all those big and little problems of Life (and of Undeath).
A little problem is with adventurers:
Gold? Too few valuables.
Killing monsters? YOU are a monster.
Fame? Very funny, my friend.
Girls? There are some, but would you really want...
What remains is the urge to do
something, just anything, not having the boring life like everyone else. It was one of the reasons most adventurers started. Now, even former farmers may want a change.
Social-being.Undead are not social. Distrust is the most common feeling if you meet someone unknown. Many simply ignore you, unless there is some reason to hear.
To sum up Charisma:Charisma is based mainly on physical fitness, and how preserved the Undead looks. A missing limb, or many wounds (even stitched together) lower it. Whatever makes the Undead look more able to survive Undead World, improves his Charisma. Thus, weapons do help in more ways. A quality sword makes everyone listen to you. A preserved zombie with good rag-armor and an old sword can have the Charisma of a paladin. (Ironic if it was a paladin before.) Having a lot of animals/servants improves Charisma, too.
Still, most Undead value their freedom and peace, and if you are known to them as a breaker of these things, all things that improved your Charisma lower it now instead.