Adjective. Taking coolness to its extreme and generally beyond the realm of actual possibility, while at the same time seemingly unaware of how ridiculous it is.
Distinctions from similar words:
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Mary-Sue: This is distinct from mary-sue because characters generally have "flaws", but which end up making them cooler as opposed to actually flawed.
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Try-hard: Edgy characters go for diversity in their coolness instead of a specific theme. An edgy character is much more diversely cool, whereas a try-hard goes for something specific, like "only using knives" or "hates color because he/she is goth".
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Campy: Edgy things are either unaware of their ridiculousness or actively try to explain it away. In contrast, campy things are totally aware of how ridiculous they are and actively embrace it.