Achilles has plenty for Connor to do. Here's a rundown:
Mission - Availability - Description
Encyclopedia of the Common Man - Sequence 06 - Begin this mission to continue the Achilles missions listed below (you don't have to complete it to finish the Achilles missions)
Manor Mysteries 1 - Sequence 07 - Achilles will be in front of the fireplace in the Manor. Follow him and then talk to Norris in the mine. Follow him to the entrance, blow up the explosive barrel with your pistol, and follow the waypoint markers. Climb up the wall and run to the first chest. After this you have two minutes to escape. Do this by climbing the wall on your right and follow the same path backward, making a leftwards path when you reach branching paths. Deliver the chest's contents to Achilles.
Manor Mysteries II - Sequence 09 - Talk to Achilles and then head to the ruined part of New York in the west of the city. The painting resides at the top of one of the ruined buildings. Once you've found it, return to the Homestead and Achilles.
Legacy - Sequence 11 - Achilles is in the manor. After speaking with him, talk to Father Timothy in the chapel.
Achilles Painting - Sequence 11 - Examine the marked wall inside the manor to complete the Achilles quests.