Hawkwood is a desolate man who stays in the Firelink Shrine. You meet him at the beginning of the game, when you first arrive at the shrine. He likes to hang out near the thrones, sitting with his legs kicked over the side of a ledge.
Speak with Hawkwood after defeating the Greatwood or Crystal Sage bosses to acquire a Heavy Gem. Defeat the Abyss Watchers and speak with him to acquire the Farron Ring.
You may occasionally find Hawkwood paying his respects to a grave outside the shrine on the right side…
After you have exhausted Hawkwood’s dialogue and defeated all three of the above bosses, Hawkwood departs, leaving Hawkwood’s Shield at the grave. At this point, you won’t see Hawkwood for a long time. He’ll appear again as a White Phantom sign in the Consumed King’s Garden.
After that, if you find your way to Archdragon Peak, his White Phantom sign can be found not far from the third bonfire. Activating it causes Hawkwood to travel up to the shrine at the end of the trail, where he prays and vanishes.
If you follow suit and use the Path of the Dragon gesture at the shrine and acquire the Twinkling Dragon Torso Stone, Hawkwood leaves a message with Blacksmith Andre in the shrine, challenging you to a duel. If you travel to the Abyss Watcher’s room in Farron Keep and face Hawkwood, defeating him earns you the Twinkling Dragon Head Stone, but losing loses the Torso stone.