At the very least he is a very Corrupt sorcerer, if not actually undead.
I think Thulsa Doom could easily be "alive" in the Hyborian age. IIRC, in the Kull story, he disguised himself as Kuthulos. He is described as having "a bare white skull, in whose eye sockets flamed livid fire."
In the more modern-day story, "Skull Face" by Howard, the skull-faced character goes under the name of Kathulos (similar to Kuthulos, yes?). The skull-faced villain later insists he is not from Egypt, but from Atlantis... He is also described as having livid fire for eyes. Skull-face, IIRC, does have parchment-thin skin, but still is essentially a skull. Still, I think a connection could be drawn if one wants to do it. It certainly wouldn't be the first time a relic from one of his past-history series appeared in his more modern tales (Thoth-amon's ring reappears in Howard's "The Haunter of the Ring," for example).
The neat thing about connecting "Skull-Face" with Thulsa Doom is that it allows more insight into his sorcery, his personality and how he is so long-lived.
Of course, Howard also had an affinity for reusing names he liked, even if the characters were not related, so this idea isn't exactly iron-clad - how many times did he use Almuric/Amalric to represent different characters?