How would you define and design a villain?
I would start by determining what I want the villain for. The speed-bump henchman of the big bad is a villain in his own right, but he's a very different proposition than his boss. So, I'll start by fixing that.
From there, I work out what the villain wants, what he's willing to do to get it, and his preferred methods. (Again, a Wizard is rather different from a Fighter, even if they're both striving to end the world!)
Once I've got that, I'll start building the villain exactly as I would a PC, although I may well play a bit (or a lot) fast-and-loose with the RAW when doing so.
But one very important consideration - I try to make sure that those villains who are supposed to be significant (that is, the BBEG and his main henchmen) are powerful enough to stand up to the combined might of the PCs for 4-6 rounds. Because without that, he's a speed-bump and won't generate any good after-action stories.
(Of course, when we get to actual play, if the players pull out some tactic or power that I hadn't expected, or just get some lucky rolls, then I'll make sure to let that stand. I may have planned for an epic battle, but if it doesn't happen then so be it - I can always get another bad guy.

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What's the difference between the abrasive anto-hero and a real villain?
Boundaries. They don't even need to be stated, but there has to be a point somewhere along the line where the anti-hero won't do something, but the true villain will. For example, Batman has no problem beating up the bad guys in order to get information, but he stops short of killing them.
How can a player portray a Chaotic Neutral PC without teetering into villain territory?
As noted up-thread, by choosing not to be a jerk. It's also worth noting that you don't need to be CN to cause problems. Inspector Javert (Les Miserables) is pretty much the textbook example of a Lawful Neutral character... and one played by a jerk.

Heck, in the wrong group and with the wrong player, even a goody-two-shoes LG Paladin can be a problem.