Chalk's headdress provides a DT of 1 and an increase of 5 to Melee Weapons. It appears to be a heavily decorated baseball cap (possibly a scout cap) with a yellow W on the front and many tribal adornments, such as the feathers, beads, and patterns that are woven onto the cap. It can be repaired with Pre-War baseball caps and with most other head gear with the Jury Riggingperk.
Despite being a hat, you cannot equip eyewear while wearing Chalk's headdress.
This cannot be obtained if you choose Chaos in Zion. The only way to obtain it this way is to kill Follows-Chalk first, then killing the remaining non-player characters.
Killing Follows-Chalk sometime between Arrival at Zion and Deliverer of Sorrows will allow you to obtain two Chalk's headdresses. This will prevent the quest Chaos in Zion from triggering, allowing you to complete the main questline. Store the looted headdress before completing either Flight from Zion or Crush the White Legs, and a second headdress will be present in the chest at the conclusion of the add-on.
Edit