Drama; Tragedy
The Revenger's Tragedy is most definitely a drama, and it's most definitely a tragedy. On the drama front, it's a play, which is pretty much all it takes to land in this genre category—and this play goes one step further, playing around with the idea of drama by staging some of the action as staged entertainment within the play. It's kind of like the whole play-within-a-play idea, and this heightens the audience's awareness of their own experience watching the drama unfold.
As for the tragedy bit, well, this one's about as tragic as they come. Vindice undertakes the dramatic task of revenge, and in doing so, he's undone by it. The body count is pretty high, and it includes our hero Vindice and his brother Hippolito. A few characters are still alive at the end, like Castiza and Antonio, but the survival stats are pretty grim, and there's no true happy ending in sight. In other words, it's a bummer through and through.