How do you get explosive bullets in Dishonored?
From the Wiki:
The only way to acquire explosive bullets is by purchasing them directly from Piero or black market shops, except for one case of explosive bullets that can be found in the Dishonored mission The Light at the End.
...they can only be purchased once the protagonist has found the Small-Scale Combustion Refinement blueprint.
According to the Wiki page for blueprints, you can find the blueprint that you will need during the Return to the Tower mission in the office of General Tobias or during the Flooded District inside an abandoned rail car in Central Rudshore.
Once you have the blueprint, you can buy the Explosive-Shot upgrade for 600, and this unlocks the ability to purchase them.