This achievement is tied to the main story mission of the same name, and happens after you have completed "Reunions".
One thing to note before getting into the details is that you will need to have the Cybernetic Brain Augmenter in your possession, which you can get from a looted body during the previous mission. I had it with me so I was good, but if anyone did not pick it up, please let me know in the comments what you had to do if you didn't pick it up and I will update the solution with that path accordingly.
*** Spoiler - click to reveal ***The main goal of this mission is to talk to a Dr. Amari in Goodneighbor at The Memory Den, at the suggestion of Nick Valentine in the conversation you have with him after completing "Reunions". The reason you need the Cybernetic Brain Augmenter from Kellogg's body is that Dr. Amari can supposedly extract memories from Kellogg's brain in order to get leads as to where Shaun is, because at this point in the story, you don't have any.
Once you meet Dr. Amari in the Memory Den basement, you give her the Augmenter, which she tries to attach to Nick Valentine, since he's a Synth. But it's not going to work with him alone, so you are going to also need to get in a chair along with him to help extract Kellogg's memories. Also, there is a Robco Fun Book in the room and you can also pick up many other aids.
You will go through several memories throughout Kellogg's life in a linear path (there is no shooting nor do you take control of Kellogg at any point - although you can walk up to each person in each scene and press to get more information, narrated by Kellogg) from childhood, including the memory that takes place the day your wife was killed and Shaun was taken in Vault 111. The final memory involves a Courser (a Synth). The Courser takes Shaun from Kellogg, and it is also revealed that the reason no one can find out where the Institute is is because the only way you can get there is via teleporter.
After all of the memories have been extracted, you will need to go see a rogue scientist (Virgil, who is a Super Mutant) in The Glowing Sea, which is in the lower left of the Commonwealth map.
Once you have completed the session and spoken to Dr. Amari again, you will meet Nick upstairs, and the first time you speak to him, he will be speaking in Kellogg's voice.
If there is anything I can do to improve this guide, please let me know in the comments.