This quest begins a trilogy of Valentine Detective Agency cases. Head to Diamond City and visit Ellie at the detective agency. She'll mention some unsolved cases. To start the first one, pick up the Earl Sterling file on or near one of the filing cabinets.
Once the quest begins, follow the waypoint to Earl's home in Diamond City. Lockpicking his Novice lock shouldn't be a problem, but if you'd rather enter with a key, visit Vadim Bobrov at the Dugout Inn.
However you decide to get inside, search near Earl's couch after you enter. You'll find evidence pointing to a facial reconstructive surgeon who works at the Mega Surgery Center. That's your next destination, which is another short walk within Diamond City.
Once you're there, speak to Doctor Sun. Ignore all the off-topic conversation choices and focus on questions about Earl's disappearance. After those choices have been exhausted, look at the ground and note the bloody marks leading to a cellar. Ask Sun about these marks. Depending on your charisma level, you can threaten or politely ask him to open the cellar door. If neither of those options work, you can try to pick the lock or bribe Sun for the key.
Once you're in the cellar, you'll find the doctor responsible for Earl's disappearance, That leads to a stand-off. You can try to persuade him to stand down, you can try to arrest him — or you can kill him. Whatever you decide, return to Ellie at the detective agency to close the case to receive your reward.