Complete the quests 'Praxilla's Admirer', 'Lyre Lyre', 'Repairing the Lyre', and 'Praxilla's Legacy' to complete the DLC and earn the achievement.
The first quest is found in the Mount Helikon region of Boeotia.
*** Spoiler - click to reveal ***You will get the quest 'Praxilla's Admirer' from Alkibiades near Port of Kreusis in the Mount Helikon region of Boeotia, closest viewpoint is Standing Pillar.
You will be tasked by Orchimedes with getting 5 Coral Rose which are located north of the quest giver on top of the hill. At the top of the hill you will find 3 wolves and the 5 roses. Kill the wolves, gather the roses and return to Orchimedes. He will ask you to deliver the roses to Praxilla for him. Deliver the roses and after some dialogue Praxilla's Lyre will be taken and you will be tasked with retrieving it.
This will close out 'Praxilla's Admirer' and the quest 'Lyre Lyre' will begin.
Talk to the people in camp for clues and you will be told to search the Temple of Athena Itonia in the northwestern part of Mount Helikon. You will find the servant girl on the road south of the temple and she will tell you Orchimedes was taken by the bandits and you will have to free him. Escort him out of the area and talk to him. Next return to Praxilla to complete the quest.
To get the next quest talk to Praxilla and you will receive the quest 'Repairing the Lyre'. You will have to escort Praxilla to Port of Kreusis. Once you arrive at your destination talk to Praxilla again and you will be tasked with getting 3 wolf intestines which can be found just to the west at Narkissos. Next you will need to gather 3 conch shells which are directly south. The next itm is Silver Birch which is east of Port of Kreusis. When you have the required items return to Praxilla to complete 'Repairing the Lyre'.
Next return to Praxilla in camp to begin 'Praxilla's Legacy' escort Praxilla to the stage east of Port of Kreusis. Upon arrival talk to Praxilla and she will ask you to recite her poem, the correct lines are Sun, Moon and cucumber. When you finish the crowd will be attacked by thugs, defeat them and then talk to Praxilla again to complete the quest.