You automatically unlock this questline after the Kings of Sparta quest.
Here are the quests:
1: Delivering a Champion
*** Spoiler - click to reveal ***Talk to a trainer about making Testikles Sparta's champion. Now, talk to the oil maker's apprentice.
Choose "I can pay you" followed by "Can't we work something out?"
Next, "First, light a fire," followed by "Oil of Athena" and then "Cinnamon".
Now, talk to Testikles, who, uh, needs to lay off the booze, apparently. Say "I've got your oil" and then "You'll get your oil in Elis".
Now, sail to Elis. If you've been there, you can just fast travel there and call your ship (Credit: Elitdarklord1).
2: The Contender
*** Spoiler - click to reveal ***Talk to the Pankration Organizer. Choose either dialog option, then "Let's go."
Win both fights without the use of your abilities. After the fights and scenes, a Cult of the Kosmos member gets revealed.
3: The Long Game
*** Spoiler - click to reveal ***Talk to someone at the bankuet. Now, find out what happened to Alkibiades. Go to the location and start searching.There are three places to interact with: The note in the back room, the table in the middle of the room, and the poison by the close door.
Talk to the merchant. Be ready for an ambush when you get there! Now talk to him. Go to Fort Korolbos and get the antidote. Finally, bring it to Alkibiades.
4: Pankration
*** Spoiler - click to reveal ***Talk to Palos. Next, win your fight with no abilities. The Olympic Champion tends to dodge. Thankfully, you can still use your overpower ability.
After the celebration scene, enjoy your "Going for Gold" achievement.