Mm, this is a pretty subjective matter.
I'd say the biggest definitions of a hardcore gamer and how often one plays and how obsessed one is, as those two go hand-in-hand.
What about skill? Well, consider a hardcore RPG gamer. Skill isn't really necessary, but they can still have a real passion.
Of course, that might just be an exception. Then, consider this. This person is a true natural at video games. But it doesn't necessarily mean he likes games, and it would furthermore mean that he, of all people, would have the least reason to play/practice often.
More importantly, skill is often a byproduct of how often we play.
Actually, how often we play is very often a byproduct of how into the game we are. So... Maybe this means that obsession is the biggest factor, whether it be in the form of how often one discusses games, how focused one is at completing games 100%, or how competitive one is.
Of course, how competitive you are can be a byproduct of how skilled you are in the first place...
Bah, I don't know. I, for one, do not consider myself a hardcore gamer, so I'm not very sure. Skill, how long one plays, and obsession are the key points, but I don't think I'll bother any more trying to figure out which one is the most important.