Preheat the oven to 425 degrees F. Cook the bacon in a skillet over medium heat until crisp. Drain on paper towels and reserve 2 tablespoons drippings. Finely chop the bacon.
Whisk the flour, baking powder, baking soda, salt, chopped bacon and chives in a medium bowl. Using a pastry blender or your fingers, work the butter into the flour mixture until it is in pea-size pieces. Stir in the cheese with a wooden spoon. Add the buttermilk and reserved bacon drippings and gently mix until just moistened. The dough will be loose.
Turn the dough out onto a lightly floured surface and knead 3 to 5 times, just until it comes together. (Don't overknead or the biscuits will be tough.) Roll out to 1/2 inch thick, then cut out biscuits using a floured 2-inch-round cutter. Arrange on an ungreased baking sheet. Reroll the scraps and cut out more biscuits. Brush the tops with melted butter.
Bake the biscuits until golden brown, 12 to 15 minutes. Serve warm.