Generally yes. Some more so than others.
Nai Calus was my first PC, a half-elven Bard in 3.5. He has entirely too much of a backstory in every incarnation of him I've done. The character made his first appearance as a Phantasy Star Online character of mine, reappeared with some alterations in Phantasy Star Universe(I can only stay involved in online games if I'm fond of my characters as characters, so they've always gotten backstories and personalities), and wound up with even more heavy background alterations as my first D&D character.
I've used him twice in 3.5, the second time in a campaign that featured one of the party members from that first campaign as one of the antagonists. He went by Celenden Theleril and was a Bard/Swashbuckler, but was still the same character and the two still knew each other. He's most recently been an eladrin cleric of Corellon named Vel Theryn in a PbP game.
No, I'm still not done with him.
(Probably because of campaigns falling apart and him being incompatible with the 'badass' CN/CE/LN leaning towards E characters people kept making and introducing when they got bored with their original characters that actually were compatible with him. The 3.5 group were jerks and I really should have dumped. I just want to play him to some sort of reasonable conclusion in a game once, damn it.)
I find that my grasp of the setting or lack of setting helps. In the little-defined homebrews of the 3.5 group it was a lot easier for me to come up with a background without feeling that it was pretentious or wrong. I like my Shadowrun character but I'm not as comfortable with him because I don't know the setting perfectly, so there's a lot of his past that would normally be fleshed out that isn't.
Creation mechanics help too. Rolled stats and rolling for things on tables isn't my preferred way of doing it at all and I tend not to care much for characters generated that way. I make the character, and then I figure out what he'd be mechanically.
Having elves in a setting helps too. I like elves. (The DM let me get away with playing an elf in d20 Deadlands once, rofl. Li Shang, 'Chinese' antiques dealer. Take your standard D&D elven ranger, strip him of his abilities and throw him into America in another universe via planar travel gone horribly wrong, figure out how he's going to hide himself and try to adapt. I remember him fondly because it was such a ridiculous character concept but fun to play.)
I've also go several PCs I'm fond of that I've never yet gotten to actually play. Maybe someday I'll get to play them. Too many characters, too few campaigns.
But yes, I do tend to care about my PCs. I'm there to pretend to be an elf and maybe kill some monsters while I'm at it. If I don't like my PC as a person, what's the point? If I just wanna kill things and take their stuff I can play video games and not deal with fiddy bits like character sheets and figuring out how to make something work mechanically.
