An Alchemist's Apparel is the second quest assigned by Barbatos at the Battle Tower's Alchemy Garden. You must have completed Borrowed Book to receive this quest, however you can fulfill the objective prior to obtaining the quest and accept & turn it in afterwards.
Barbatos requests that your retrieve the Alchemy Amulet from Rayhun, a Black Ring lieutenant. He can be found in the Broken Valley where the village used to be. Fly to the area using your dragon form, taking out any air defenses as needed along the way. On the ground, search for a green teleporter - the entrance to Rayhun's Headquarters.
The headquarters is small. Simply follow the halls until you find Rayhun, slaughter him, and take the amulet. You may want to open the nearby chest for the Aleroth Archmage Helmet and Crystal Dragon Tail Piece while you're here.
Return to Barbatos to give him the amulet at your convenience.