To unlock the A Friend Worth Dying For achievement in Assassin's Creed Odyssey, you need to have Update 1.1.4 installed and have completed at least Chapter 5 of the main quest.
The first quest of this arc, "Odyssey into the Past", can be found in the Plains of Achilles region on the island of Skyros. The closest viewpoint is the Temple of Achilles. The quest will be highlighted on the map with a blue diamond with an exclamation mark in the middle.
Odyssey into the Past
The first quest marker is located in a Pirate Hideout. The pirates are all drugged, so there is no need to fight. Free the prisoner and talk to her. After the cutscene, you will need to kill the bandits in the area and then talk to Barnabas. After some loading, you will appear on the Adrestia with Barnabas. Follow the dialogue choices and you will unlock three new quests.
Beware the Siren Call
The quest marker can be found in the Siren's Tomb in the Pharmakeia Bay region of Kos. The closest viewpoint is the Alter of Apollo. Smash the door at the entrance to gain entry to the tomb, walk in and a short cutscene will play. make your way through the tomb, killing the few enemies along the path. In the final room, you will have have to kill the siren and loot her. Once completed, there is a short cutscene and some dialogue. Choose whichever options you wish. When you have finished talking, the quest will be completed.
To Be Nobody
The quest marker can be found in the Drakontospilo cave in the Ash Hills region of Anaphi. The closest viewpoint is Mount Bigla. Follow the linear path through the cave until you trigger a cutscene and some dialogue choices. Follow Polyphemos, defending him along the way. Once you kill the final enemies, talk to him and follow the dialogue choices however you wish. Once completed the quest will be over. (I only chose the kinder options, which ended with me receiving the quest items. You may have to kill/loot the items if you choose differently).
Wine for the Swine
The quest marker can be found in the Alkaios Tomb in the Atoll Point region of Paros. The closest viewpoint is Athena's Watch. Talk to the witch and you will have the to choose to fight her or drink the wine. You can choose whichever option you wish. You will receive the bracelet that you need to complete the quest whichever way you decide.
Once you have completed all three quests, return to the Adrestia and talk to Barnabas. Go through the dialogue choices and choose whichever options you wish to complete the Odyssey into the Past quest.
The quest marker can be found at The Entrance to the Underworld in the Parnes Mountains region of Attika. The nearest viewpoint is the Abandoned Watchtower in the Ruin of Plataia region to the northwest of the quest marker. Dive down into the Entrance to the Underworld and follow the path. There will be trails of blood on the floor to indicate the way you need to go. Once you reach the cutscene with Barnabas in the cave, you will meet with Leda.
You will be given two options. You can choose whichever you wish. I chose the option to deal with Leda's suitors and had to follow her and then kill the two suitors. Once finished, talk to Leda again and choose the options you wish to complete the quest and unlock the A Friend Worth Dying For achievement.