Assuming you were thorough in looting Kellogg's corpse, you'll have a piece of Kellogg's brain with you. The next step is to find a way to analyze Kellogg's memories for a clue on the whereabouts of Shaun. To do that, you need to go on a brief trek to meet a woman named Amari in Goodneighbor in the east, close to Park Street Station. Simply follow the waypoint indicator for the precise location. Expect a couple mutants along the way.
When inside Goodneighbor, look for Memory Den, which is Amari's place of work. When you meet her in her lab, be sure to take your time looking around. You'll find a lot of aid items and even a holotape game, Grognak The Barbarian, a bonus item from a RobCo Fun magazine. Converse with Amari about your missing son and convey your desperation in finding him. While using Nick as a memory catalyst, you yourself will be diving into Kellogg's memories.
When you've jumped into this dreamlike memory state, follow the neural paths connecting Kellogg's many memories. It's a very linear sequence, eventually leading you to the answers you're looking for. After the memory dive, talk to Amari to complete the chapter.