How can a Lich NPC be made more believable? Things like...
Tim's answer is great, but I want to expand on it. This especially applies if you're thinking "NPC villain" and not just "powerful monster."
Start by remembering that a lich was a person... human or not, it was an intelligent creature that, for some reason, went to great lengths to cheat death. When you think of your lich as "person with some serious obsession and lots of power," he can become a much more interesting villain.
This ties into Tim's "obsession" issue... why did he cheat death? What was so important to him that he expended so much of his time and resources to become an undead monster? Maybe just fear of death... some reason he doesn't want to cross over to the other side. Considering that in typical fantasy RPGs, "the other side" is mostly a known quantity, what horrible fate did this guy expect to receive? What did he do that guaranteed such a miserable afterlife that he'd rather continue as an undead than move on? If that's all he's really afraid of... what does he do now that he's secured his safety from his biggest fear? Imagine a lich, bored to tears (if he had any), with nothing to obsess over but the safety of his phylactery. This might be the kind of lich that just "sleeps" for centuries at a time.
But maybe death was just an obstacle to getting something done... again, Tim's "obsession." But obsession doesn't have to be a journey ("learn moar magics!")... this lich's obsession could have a destination in mind. What thing did he need to accomplish so badly that he was willing to cheat death and become a monster to achieve his goal? Imagine the lich who is really just a loving husband, who has spent 1000 years trying to find a way to bring his lost wife back from a fate worse than death. Imagine the reunion when he finally succeeds and she sees what he has become.
It doesn't have to be so romantic, of course... maybe it's just power. What's the point of being king if dying means you lose all your toys? The lich king is a common theme... instead of holed up in a tower, this lich sits the throne and runs his kingdom with a dry, cold fist of iron. Which means, in additional to spending those sleepless nights researching more magic, he's got a kingdom to run and his day is much like that of any other tyrant. And even though he knows he can't be killed, imagine his worry about what happens to his kingdom if he disappears for a week while his body regenerates?
Regardless of why he did it, this former person, had a reason to cheat death... and once that pesky death issue is out of they way, he's going to keep right on pursuing his original goal. He's just as intelligent as he was before, just as resourceful. Imagine that he'd simply gained immortality instead of becoming undead... how would his plans proceed? He's not going to enjoy a normal life with the wife and kids (being undead is hard on relationships, but then so is his level of obsession), but isn't the lich going to be doing pretty much the same thing as if he were simply immortal?
Aside from the guy whose end goal is cheating death, think of lichdom as just one milestone in this guy's overall plan. Put yourself in his shoes and ask, "What do I do now to get closer to my goal?" And if he's ancient and still hasn't reached his goal, he's probably bat-guano insane, so there's that to play with.
You don't mention a setting, and that could make a big difference. For AD&D 2e (Ravenloft specifically) there was a sourcebook entirely on Liches that at least touched on this. I believe it was Van Richten's Guide to the Lich.
But let's look at what we might call a "stereotypical lich." Stereotypically, Liches are obsessive. You tend to need an obsessive personality to do the things necessary to become a lich, and some settings assume that unlife further heightens this.
Being obsessive and being freed of most bodily functions, the Lich will spend his time doing things that relate to his obsession. The stereotypical obsession is with magic itself and the lich could spend all of eternity studying. Possibly rarely leaving its lair as it reads, experiments, enchants, writes, etc. It may of course leave its lair to gather things that will help it, such as finding new tomes, new items, and more equipment.
Of course, even if the obsession is with "magic" (and steroetypically it will be related to magic somehow), it could have plenty of nuances. Perhaps the lich is obsessed with learning through teaching and takes apprentices on a regular basis. Or it might be obsessed with a particular type of magic which requires it to go forth and gather specific components frequently. Or perhaps it hoards and collects a particular type of magic item. All of these would add flavor to a particular lich while also giving reasons for it to interact with the outside world and come to the attention of adventurers.
