Well I personally think you just need to really understand the character. One thing that annoys me with vampire characters is that they're just too cliche now! Society has turned away from the classic souless monsters and gone into the 'emotional' immortals. Going to use Death as an actual main character in my next short story after I finish what I'm currently writing. Death basically decides to stop someone's demise, breaking it's own rules, and wonders why it is that they did this. To actually write this story I need a perception of what an immortal/ancient-being would be like. Many people have written these in different ways, so you just need to capture how you want the person to be. Immortals are fine, giving them a weakness is fine too. It's really going to come down to how you write it though that will make or break what you do. I can't guide you as I have no idea as to the context of the character you want to use, but remember: 'Everything Has Been Done' = You need to make it unique.
An immortal character is an interesting idea. You should make sure they are not also invincible though. If they can't be defeated, or denied their goals in some way the story would be rather boring.
There are ways to make immortal characters interesting, but they need to have depth, obviously. Take Dr Manhattan or something- what makes him interesting is that despite being invulnerable to everything, by about halfway he sees no more reason to save his fellow man- he has no reason to save his planet. Within his character you see elements of his past life and you see how divorced he has become from it. Superman, on the other hand, is an example of an invulnerable character done poorly. He is immune to everything (except kryptonite, yeah, okay), has every power ever and is incurably honest, noble and has almost no conflict. He is not believable as a character because he never does anything wrong. Just because a character is immortal it doesn't mean that they should be made any less human.
Baldr was no more immortal than Achilles, Macbeth or Superman (ok, as long as he stays under a yellow sun). There's all the difference in the world between being killable in zero ways and being killable in one way.
The semantics are indistinguishable until the supposed immortal finds his or her vulnerability. And given that the lifetime of the universe is itself finite, then there can be no true immortals. In the end, time will triumph.
The term "immortal" seems to have arisen in the context of religion. The traditional Christian conception of God ("Immortal, invisible, God only wise...), for instance, is considered to transcend the universe and even time. This probably isn't the place to debate whether such an entity can or does exist (or even whether, as Paul Tillich suggested, to talk about God "existing" is a category error). But to call an entity that is bound by time "immortal" is to devalue the word.
