I'm pretty sure you can see it happening if you watch like a hawk. But it can take a while, and you don't get anything else done while you watch.
Another good way to narrow the field down a lot if go through and look at their happy thoughts. A vampire will never have thoughts about dining in a legendary dining room, having a great meal or a great drink. This is pretty painless to do using Dwarf Therapist. I just go down the list and hover over each dwarf to get the popup with their happy/sad thoughts. Any that don't have dining/drinking thoughts get their alchemy skill turned on as a flag. In my current game I was able to narrow it down to 3 dwarves that way. All were in the military and had been on duty long enough that they hadn't eaten in the dining room recently, and my booze/food quality wasn't yet up high enough to reliably get the thoughts for quality.
Then I just looked at the relationships for those 3 (u->select dwarf->c->z->r). One had a wife, 12 kids, and dozens of aunts, uncles and cousins, none of whom were at the fort. He also was a former member of 10 or so groups. The other two were quite normal, both being married to spouses who are in my fort and not too many relatives. Only historical figures (like vampires) will have many off-screen relatives and former organizations.
The vampire also had some interesting traits. He dreamed of ruling the world, for one thing. I've never seen a dwarf with that particular dream before. He also had a bunch of traits like not respecting the law, not getting along well with others, and a few more that seemed very appropriate for someone who makes a living by murdering people in their sleep. I don't know if this is typical for vampires, or if this guy was just a sociopath before he became a vampire. It makes sense that if he has survived for a hundred years as a vampire that his personality would have adapted. Someone should do some ‼SCIENCE‼ by checking a dwarf's traits before and after causing them to become a vampire.
When it comes to identifying vampires, how hard it is is really only determined by how much you are willing to exploit bugs. The list of former organizations is probably a bug. I'm not sure about the list of relatives, but it probably is (they really should come up with better background stories). As people always point out, it's a single player game, so if you cheat, you're only cheating yourself. Or making the game more enjoyable. Whichever way you prefer to look at it.
Another exploity way is to look at the deity(ies) listed in the relationship screen. A vampire will have a deity who has cursed at least one person to roam the night searching for blood to drink (or something like that, but don't confuse it with the were curses). This alone may or may not be considered an exploit, but if you give the dwarf a nickname like "vampire?" and look at the list of who the deity has cursed and now see one of the cursed dwarves with that nickname, you have definitely found your vampire, and also pretty definitely exploited a bug.