Another very common element to vampires is enchantment and domination. Perhaps you could force peasants/companions/enemies to do things, mesmerize them, tell them to forget that suspicious blood stain on my collar, Have them fight for my amusement, ect. Related is seduction. Vampires are often sexual deviants, from homosexual tendencies to the stealing (and corruption) of pure young maidens. all That would be a very complex and time-consuming mechanic though.
Something that is very obviously missing is regeneration for vampires.
Another is the ability to consciously convert your victims into more vampires, often the new vampire is dominated and the slave if their creator. Thus one can have dens of vampires who either follow a leader or scheme against eachother.
The ability to turn into a swarm of bats, mist, ect, and fly for a short time would be neat.
Needing to return to a graveyard or coffin would be neat if coffins and graveyards were actually a thing in adventure mode. Better would be being destroyed or weakened by the sun. You are a NIGHT creature after all.
Different ways to discover that you have become a vampire have allready been mentioned.
Finally vampires should be harder to kill (even withou regeneration) certain types of vampires should come back unless a particular condition is met. Killed by a weapon of a particular material (including ash stake) buried or disposed of a certain way, burning the body, decapitating the body, burying it on holy ground (perhaps of the Deity that originated the curse) killed durig a certain time of day or during a certain phase of the moon. The possibilities are endless, and finding out what a particular vampire's weakness is should be difficult and interesting on it's own right. Like finding a clue at the location where they were originally curse. Reading notes left behind by past vampire slayers. NOT by asking random peasants for random trivia. Finding a vampire and discovering a vampire's weakness should be worthy of a quest in its own right. And there should be penalties for getting it wrong, the most obvious being that it rises again and now is angry and chasing you down, or perhaps has taken a new identity and is biding its time.
Mixing and matching these would really make vampires more fun to kill and to play as. Most of these examples are difficult and time-consuming to implement though and most require other changes the most obvious being handling the way conversation and supply of information is handled. A vampire can't hide its identity in the current system which should be half the fun of tracking one down and significant fun in playing one. If every peasant in town can point you towards the nearest vampire and they announce themselves boldly at first provocation... Well I realize they are incomplete, but there's lots of possibilities to be had. Peasants could well intentionally point you in the wrong direction. The suspect may be a scapegoat or the subject of a witch hunt. And obviously a vampire should profess his innocence unless he is cornered by overwhelming evidence againts him. In which case he might, instead of fighting, turn into mist and flee to his lair. Where if you hadn't done your research he will be completely safe from you. I think procedureally created vampire would really lend themselves well to this type of investigative play.
Of course I don't see any of this happening any time soon. Perhaps the only realistic suggestion for now is that vampires should get some degree of regenerative abilities, and some degree of weakness to daylight.