For ghasts, ghouls, and vampires (and other fairly intelligent undead) I see most of them existing in a state similar to Henry's explanation with something additional: hunger.
Hunger drives a ghoul, and it's never truly satisfied. A deep body-aching hunger that cannot stop, even when your mouth is full of rich, red meat. The fleeting feeling of satisfaction that Henry mentioned, which drives them to consume and consume.
A vampire, well, there are tons of different takes on vampires. Go read seven or eight vampire novels and form a rough idea from that, I'd say. For the most common D&D vampire is going to be the vampire spawn, and they're mostly like ghouls save they hunger for blood.
Your major vamipre lord, though, he's looking not for blood but for life force. The yawning void inside him wants to be filled and it will not be denied, even if he doesn't want to feed. He will do so, eventually, and that probably drives most of them mad. Or they may just like killing, the ending of life. It gives them a rush of some kind. (Notice how most of this comes back to some form of pleasure?)
For the intelligent undead, many of them will remember their former life. Some may remember it vividly, or if they've lived long enough it might be just a fleeting memory. Some ghouls might become complete animals, while others may retain little bits and chunks of memory. Vamires will probably retain most of it, and may indeed be able to continue on with their old lives if conditions are right. Liches certainly do: there's little point to becoming a lich if you forget things.
For the mindless undead, I think of their spirit being trapped in there though I'm pretty sure the MM says that nothing is there anymore, just an animating Negative force that is like a vacuum: it seeks to fill itself with positive energy and thus it seeks out the living. Since they can't energy drain, though, that void is never filled and so the vessel goes on trying to fill itself. The horrific relentlessness of the shambling zombies in Night of the Living Dead should be the guide here. There is no mind, no caring: it's just a machine on two feet that exists to eat.
Incoporeal undead must have a very bizarre existance. I think Henry's 'need to feel something' idea works best with them, since they've lost the most. Wraiths and specrres especially.
