In order to obtain this Trophy your character needs to gain enough experience to progress to Level 50.
Completing quests is a good way of gaining Exp, significantly the repeating Minutemen Quests set by Preston Garvey. If you travel to Sanctuary Hills after he and the other settlers move in, Preston will continually issue quests to set up new settlements and defend old ones. Each quests takes between 10-15 minutes and will award you a generous amount of Exp, taking you to Level 10 reasonably quickly.
Alternatively exploring the Commonwealth and adding locations to your Pip-Boy will award a small amount of Experience. This milestone will take a considerable amount of time to reach, mainky due to the way progression works in this game. Each time you reach a new level it will take a larger amount of experience for the next level.