The prerequisites for this heartfelt quest are as follows:
Spend enough time with Nick Valentine as your partner to the point where you've gained some affinity with him.
Have Nick Valentine as your current partner.
With those requirements met, Nick will tell you the story of a cold case close to his heart — or to be more accurate, close to the heart of the original Nick Valentine. He's in search of Eddie Winter, an infamous Boston gangster who murdered Nick's girlfriend 200 years ago.
Your first objective is a long one: You need to travel all over Greater Boston and The Commonwealth to nine police stations and recover nine holotapes pertaining to Eddie. Nick will already have one tape for you, and marking the quest on your Pip-Boy is a huge help. Not all the waypoints will appear at the same time, but as you recover each tape, new waypoints will appear until you find all 10. And even though all the tapes are numbered, there's no proper order in which you should pick these up.
No single location is especially difficult to access. Some are completely vacant while others will be inhabited by Raiders or Super Mutants. With every police station you visit, your first step will always involve accessing a terminal and reading the file pertaining to Eddie Winter. Doing so will make the holotape appear near your current location, usually just a couple rooms away.
When you have all the tapes, hand them over to Nick. He'll use the tapes to figure out Eddie's location, which is in Andrew Station. Once you're there, take out the local Raiders and follow the waypoint to Eddie. The tunnels and sewer passages can be a bit confusing, so be sure to search around for all possible paths. Your destination is behind a metal door with a keypad. Nick recently gave you a code for the keypad. Enter it to open the door.
When you confront Eddie, your only choice will be to kill him, so making small talk is optional. Once you kill him, exit the station using the nearby ladder leading to Joe Spuckies Coffee Shop.
Your final objective is to follow Nick to a sentimental spot. Then you'll have the option to have a heart-to-heart. With this quest complete, you'll be able to earn the Close To Metal perk, provided you've maxed out your affinity with Nick.