To unlock The Heir of Memories trophy in Assassin's Creed Odyssey, you need to have Update 1.2.0 installed and have completed at least Chapter 5 of the main quest.
The first quest of this arc, "The Heir of Memories", can be found at the Temple of Poseidon on the small island in between Chios and Samos (far right of the map). You will have to sail to the island to begin the quest, the closest viewpoint with a port nearby to board the Adrestia is either Leos's Vista in the Forest of Tears region of Chios or the Rocks of Krenes in the Persian Bay region of Samos. You will have to sail to the small island regardless,
The Heir of Memories
Travel to the small island, talk to Aletheia and follow the conversation choices. After a short loading time, you will return to the present day as Layla. You will be directed to go to the seal. Walk over and following a cutscene, you will be returned to the past playing as Alexios/Kassandra. Follow the conversation and you'll finish the quest and unlock the next one.
The quest marker can be found on the island of Seriphos. The closest viewpoint is the Temple of Athena on the same island. Travel to the location and you'll find Theras, Dispose the guards choosing one of the three options available and then talk to Theras. Follow the conversation to complete the quest and unlock three new quests under the Three Symbols Entombed tree.
The quest marker can be found in Agamemnon's Tomb in the Palace of Agamemnon region of Argolis. The closest viewpoint is Heraion of Argos in the Hera's Watch region of Argolis (south of the quest marker). Enter the tomb and talk to Aletheia. After a short loading screen, you will return to the present day.
Follow the natural path. In the room with the spikes on the floor, you will have to move a sliding pillar to be able to jump over the path and progress. In the next room when you hear Aletheia say the first symbol is near, smash the pots on the left of the room and slide under the wall. The symbol is on the wall as you pass through and looks like three stars. Interact with the symbol to complete the quest and return to the past as Alexios/Kassandra.
The quest marker can be found in the Tomb of Eteokles in the Sacred Plain of Demeter region of Attika. The closest viewpoint is Abandoned Watchtower in the Ruins of Plataia region of Attika (north of the quest marker). Enter the tomb and talk to Aletheia. After a short loading screen, you will return to the present day.
Follow the natural path down, continue past the tomb until you get to a room where you can't make any more progress. Smash the pots on the left, walk around and move that pillar, walk back around and there is another pillar that you can move and push into the space made from the other pillar. When it is moved, a hole in the wall will be revealed. Go through the hole and continue down the path to find the next symbol that looks like two rings on top of an altar. Interact with the symbol to complete the quest and return to the past as Alexios/Kassandra.
The quest marker can be found in the Tomb of Orion in the Orophos Heights region of Boeotia. The closest viewpoint is Kadmeia in the Cursed Land of Oedipous region of Boeotia (west of the quest marker). Enter the tomb and talk to Aletheia. After a short loading screen, you will return to the present day.
Follow the path to the left and watch out for spikes on the floor. When you reach the first big room on the right, continue through it and carry on following the path. In the next big room, smash the pots and pass under through the hole in the wall. Eventually, you will come to a room with water on the floor. Above the floor is a rope. Move the sliding pillar at the back of the room so that you can jump up on the rope, climb to the floor above and the next symbol that looks like a tree is on the wall. Interact with the symbol to complete the quest and return to the past as Alexios/Kassandra.
Three Symbols Entombed
After another loading screen, you will take part in a trippy cutscene where you'll have to make a choice. Once it finished (and more loading) you will return to the present day again. Return to the seal again and take part in a cutscene where you will have to name the symbols.
Choose the following order:
After another cutscene, The Heir of Memories trophy will unlock and you'll return back to the past as Alexios/Kassandra.