Gringoire is bitter because the world has yet to recognize his talents as a playwright, but there's a good reason for that: he doesn't actually have any. Gringoire is the only person in the world who actually thinks he's any good, and the fact that no one else is able to appreciate him for the towering genius he thinks he is makes him incredibly bitter.
Like a lot of people who've failed in life, Gringoire is neither willing nor able to do anything constructive. Instead, he wants to destroy, to get his own back on a world that he believes has so cruelly rejected him and his art. This explains why he's so keen to arrange an attack on Notre Dame: in attacking the church, he'll be destroying a hated symbol of a society that has rejected and impoverished him. What's more, getting involved in such an act of wanton destruction will make him feel like somebody instead of the bitter, twisted nobody that he is.
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