How to unlock: You must complete the When Freedom Calls main quest and meet with Mama Murphy in Sanctuary (M1,3).
This quest won't appear in your journal as a normal mission, but as a series of small tasks. Wait until the Minutemen arrive in Sanctuary and meet with Mama Murphy in this location. The old lady will ask you to bring her a drug called Jet. You may agree to become her drug dealer (some of your companions will dislike that!) or you can turn her down. If you've chosen the first option then you may begin searching for Jet (it is a common object found in the world) or you can buy it from one of the merchants (visit Diamond City or, for example, first meet with Wolfgang in Drumlin Restaurant). Return with Jet to Mama Murphy and give it to her. She will have a new vision and you'll receive experience points.
Wait a few hours and meet with Mama Murphy again. This time the woman will ask you to build her a unique chair. Open the workshop window, choose the Special tab and then Mama Murphy's Chair. Place the chair inside the building, somewhere in the area near the old lady. Mama Murphy will be tired so once again you'll have to wait a few hours before talking to her.
The next part of this quest has a rather simple structure. You can continue delivering Mama Murphy more drugs. She will ask for Mentats, Med-X, Buffout and Psycho. Each time you give her a new drug she will have a vision and you'll get some xp. Hold off with giving her the last drug - Psycho. The reason for that is because Psycho will turn out to be too strong for her organism and she will die immediately after taking the drug.
Quest rewards: Experience points.