Access the rooftops by either climbing the ladder near The Black Cat, or jumping onto a roof from one of the castle walls. The badge is located in a box on a balcony. Follow the quest marker to the rooftop so that you are standing on the quest marker. Walk along the north edge of the roof you are on while looking straight down over the edge. You will see a small wooden platform jutting out from the end of the balcony. Jump down to the platform and walk into the balcony. Break the box to reveal the badge.
The balcony can also be reached easily using the Levitate skill or performing a Double Vault from a running start.