There are two things that you need to do to be able to get access to the Headmaster’s upper study.

The first is to complete two of the main quests, firstly “The Headmistress Speaks” and then the following quest, “The Polyjuice Plot.” This will get you into the upper echelons that is the Headmaster’s office - without doing those two quests, you’ll lack the required password to gain entry.

At this point you have already got past “Percival Rackham’s Trial” and “The Caretaker’s Lunar Lament” quest that pops up after, giving you access to the Alohomora spell. However, that will only allow you to lockpick level 1 locks.

To get into the Headmaster’s upper study, you’re going to have to unlock levels 2 and 3 of the lock picking spell. You can do that by finding 9 Demiguise statues in the open-world and then returning to Mr Moon, who will teach you the level 2 of the spell. Then, find 13 more and return to him. This will unlock the 3rd and final level.

Then, when you have access to his office and can pick level 2 and 3 locks, you’ll be able to make your way to the top, through one of the locked doors in his office. Once you’ve stepped into the room, the trophy unlocks.

The nearest fast-travel Floo Flame, by the way, is the Trophy Room in the Grand Staircase.

As an aside, don't forget to grab The Key of Admittance, next to the gramophone, while in there, as that will give you access to the locked door at the other side of the access corridor (where you first gave the password). There's 2 Collection Chests and a Revelio page inside - and it also counts as one of "Hogwarts' Secrets."