There are a number of good points in this thread, and I'd only add a couple of points that largely draw from my own experience playing villains.
Villains seldom consider themselves villains. This is the biggie. The stereotypical moustache-twirling villain knows they're a villain and they get off on it. These people, as has been pointed out, don't really exist outside of stories. Come up with a motivation for your hero that allows them to feel like the good guy. Or at least righteous.
In my opinion, there are two basic types of roleplay: Top-down and bottom-up. Top-down roleplay has a storyteller that comes up with a plot, and people play within that plot. A good example is Askier with his very large and engaging plots. Bottom-up roleplay doesn't have a storyteller, and the plot springs up out of the unplanned interactions of the characters. Most tavern RP is bottom-up, but you can get some pretty interesting stories that develop in this style, too. In the real world, I'd argue that there's no such thing as a plot that lives squarely in one category or the other. They're a blend, leaning one way or the other.
I bring these categories up because they really affect how you play and portray your villain. In a heavily top-down story, the storyteller has strong control over how things progress, and they frequently want a villain who's clearly recognizable as such for the story. These villains can be much more standard moustache-twirling - not always, of course! I mentioned Askier as an example of someone who does a lot of top-down RP. For those of you who've participated in them, ask yourself if Askier is the villain in those stories.
In bottom-up RP, you rely on your ability to interact randomly with people for RP. It's hard to do this if everyone recognizes you as a villain, so this is a circumstance where you'd want to make your villain less recognizable. Less cut-and-dried. I do a lot of bottom-up RP, so I'm pretty careful to make sure that C'kayah never stands on top of a table and yells "You shall all grovel at my feet, or my name isn't Doctor Evil!" Instead, he lives much more in the grey area. In his mind, he's the hero of his story. What's more, he's very likable. This is always a good characteristic to have in a villain. If your villain is likable, then they'll draw people's sympathy in a story (which makes things more engaging for the players), as well as stirring the pot more because they'll get so many unlikely allies ("What? No! They couldn't have done that! They're so nice! And they tip well!").
Another aspect of not making your character recognizable as a villain is simply not doing anything that convinces people you're a villain. With C'kayah, I enjoy the sort of RP where he has to dodge the law and the like. Because of this, I do crime RP out in public (in out of the way places) to allow people the chance to overhear. At the same time, I try not to discuss major crimes while sitting in the Quicksand within listening range of a bunch of Sultansworn. Because I do a lot of bottom-up RP, it becomes a balancing act. You don't want to go too far towards the secrecy direction because then no one has a chance to notice anything, but you don't want to go too far towards the public-act direction because then everyone knows you're Doctor Evil. This is one of the more fun aspects of playing a villain, because it's so immediate. I'm not simply playing C'kayah. I'm also putting myself in his head as he tries to get away with crimes without getting caught, and it's never entirely in his control!
Playing a villain is a lot of fun, and I hope you enjoy it.
Villains seldom consider themselves villains. This is the biggie. The stereotypical moustache-twirling villain knows they're a villain and they get off on it. These people, as has been pointed out, don't really exist outside of stories. Come up with a motivation for your hero that allows them to feel like the good guy. Or at least righteous.
In my opinion, there are two basic types of roleplay: Top-down and bottom-up. Top-down roleplay has a storyteller that comes up with a plot, and people play within that plot. A good example is Askier with his very large and engaging plots. Bottom-up roleplay doesn't have a storyteller, and the plot springs up out of the unplanned interactions of the characters. Most tavern RP is bottom-up, but you can get some pretty interesting stories that develop in this style, too. In the real world, I'd argue that there's no such thing as a plot that lives squarely in one category or the other. They're a blend, leaning one way or the other.
I bring these categories up because they really affect how you play and portray your villain. In a heavily top-down story, the storyteller has strong control over how things progress, and they frequently want a villain who's clearly recognizable as such for the story. These villains can be much more standard moustache-twirling - not always, of course! I mentioned Askier as an example of someone who does a lot of top-down RP. For those of you who've participated in them, ask yourself if Askier is the villain in those stories.
In bottom-up RP, you rely on your ability to interact randomly with people for RP. It's hard to do this if everyone recognizes you as a villain, so this is a circumstance where you'd want to make your villain less recognizable. Less cut-and-dried. I do a lot of bottom-up RP, so I'm pretty careful to make sure that C'kayah never stands on top of a table and yells "You shall all grovel at my feet, or my name isn't Doctor Evil!" Instead, he lives much more in the grey area. In his mind, he's the hero of his story. What's more, he's very likable. This is always a good characteristic to have in a villain. If your villain is likable, then they'll draw people's sympathy in a story (which makes things more engaging for the players), as well as stirring the pot more because they'll get so many unlikely allies ("What? No! They couldn't have done that! They're so nice! And they tip well!").
Another aspect of not making your character recognizable as a villain is simply not doing anything that convinces people you're a villain. With C'kayah, I enjoy the sort of RP where he has to dodge the law and the like. Because of this, I do crime RP out in public (in out of the way places) to allow people the chance to overhear. At the same time, I try not to discuss major crimes while sitting in the Quicksand within listening range of a bunch of Sultansworn. Because I do a lot of bottom-up RP, it becomes a balancing act. You don't want to go too far towards the secrecy direction because then no one has a chance to notice anything, but you don't want to go too far towards the public-act direction because then everyone knows you're Doctor Evil. This is one of the more fun aspects of playing a villain, because it's so immediate. I'm not simply playing C'kayah. I'm also putting myself in his head as he tries to get away with crimes without getting caught, and it's never entirely in his control!
Playing a villain is a lot of fun, and I hope you enjoy it.