Achilles is one of the inhabitants of the village. The man is involved in a few tasks that were described in the chapter below of the Assassin's Creed III Remastered guide.
You will complete the mission as you play the main story mode.
Manor Mysteries 1
Speak with Achilles and follow him to the hidden cellar. You will have to speak with Norris the miner.
The man will plant explosives by the cave and you will have to shoot them with the pistol.
Afterwards enter the unlocked tunnel and keep heading towards the designated spot. By the end you will find the chest.
Pick the lock and the cave will start crumbling. Quickly jump onto the opposite wall and keep moving left.
After reaching the tunnel, run towards the exit while sticking to the left side of the corridor. If a passage gets blocked by a falling boulder, quickly return to the last fork in the road and choose a different path. By the end, more boulders will start falling and you will have to dodge them by slipping under. You can do it by running forward with the free run button pressed. If Connor doesn't slip under the first time, back out a little and try again.
After finally getting outside, return to Achilles in the basement. That way you will end the mission.
Manor Mysteries 2
After speaking with Achilles, head to New York and find the homestead icon.
After approaching the destination, press the interaction button and a green dot marking the place where the package is hidden will appear.
To obtain it, go through the roof onto the upper floor of the burned building and open the chest.
The mission will end as you give the obtained item to Achilles. To obtain some interesting information, speak two more times with the old man in the basement.
As you enter Achilles' manor, you will find the corpse of the old man.
After a short cutscene, head to the church in the village and arrange a funeral.
The mission will end when you dig a grave for your mentor with the attack button, while listening to what he wrote in his letter.
In order to complete this mission, head to the manor and hang the painting of Achilles and his family above the fireplace.