Adachi joins the party in either late chapter 3 or early chapter 4, shortly after Ichiban and Nanba acquire a home. Shortly after joining the party, Adachi takes everyone to Survive Bar, located in the northwest part of the map, as part of the main story. Once Survive is unlocked, you unlock the ability to progress the bond level with your party members.
Adachi's bond level can be advanced by finishing battles, which typically nets 5 points, and eating meals together, which nets 10 points.
Once you get Adachi within a certain amount of points of hitting the next level, you will stop gaining points and need to go view a scene with him at Survive Bar before advancing to the next level.
Adachi's final scene to get to level 5 has a boss fight with only Ichiban and Adachi, versus level 20 enemies, so make sure you are around that level before attempting it.