The Lost Adventurer is the NPC that looks a lot like Link from The Legend of Zelda (definitely not coincidence). He will first appear in a chamber cell. If he happens to show on your run, search the entire floor for the cell key and let him out. Interact with him and look out for this adventurer on subsequent runs.
He will now occasionally and randomly appear in the Gungeon. Do not shoot at him or he will not offer his quest. Speak with him and accept his quest to explore the entire floor, apart from secret and boss rooms.
Once you have explored the entire floor (you will receive an on-screen notification once you have), speak with the adventurer to receive an item or gun for your troubles.
Repeat this process on each floor to earn the achievement. There's no way (in-game) to keep track of which floors you have mapped for the adventurer. After mapping all five chambers, speak with the Lost Adventurer in the Breach to receive the "Excalibur."