Note: As Neeker75 has pointed out, if you enter the tower in Sinclair's disguise, it will become a trespassing zone if you enter as Blake later. It is advised to kill the targets without using Sinclair's disguise, just to save the hassle later down the line.
Okay, so I'm going to start off by saying this is a bit of a process to get this secret exit to open up. It has probably the longest set up of the hidden achievements in the game.
Start off by taking out the 2 targets however you see fit, preferably without raising an alarm. Next up, you need to get a specific disguise (this step I haven't fully confirmed, so if someone can do this achievement without the disguise let me know and I will edit the solution). You need Nathaniel Blake's disguise. For reference, he has a loop that takes him from the necklace to the top floor and back again. He wears a long beige (I think that's the colour) coat and hat.
The easiest way to get his disguise is to wait until he is walking away from the necklace, then overflow the sink in the bathroom next to the double doors he goes through as he comes through them. He should inspect the distraction, so knock him out put him in the box here and take his disguise.
Now go up to the top floor, and one of the doors up here should lead to a small bridge of sorts that leads to a small building connected to the main one where the targets are. There should be a guard outside the door here, sneak right, and when he isn't looking, drop down and hang from the edge here and climb into the window on your right. In here, you should see a guard and a butler. Wait for the butler to finish speaking to the guard, then talk to him. He will now follow you.
This is the longer part of the process. In homage to Tomb Raider 2, where you could lock the butler that followed you in the freezer, you now need to do just that. Make your way back into the building where the targets were, and lead the butler out of here and down the stairs towards the phoenix effigy. Once down the stairs, on the right should be a room with a hole in the wall, where 2 guards are on the other side. Throw a coin behind them to distract them and head dead right along the wall after going through the hole to find a switch to open a hidden door.
Go through the newly opened passage (it's opposite the hole in the way when you go back through it) and go down the stairs. In this room, go dead ahead and slightly right to find another door. Go up the small staircase on the left and go right at the top. You should see a big metal door, go in here to get to the kitchen.
Go to the right in here to find another metal door leading to the freezer. Go in, and once the butler walks in, the door will lock behind him and he'll make a reference that his cousin has had the same happen to him (you'll also hear a chime). Go through the window in the room to get out and get back to the Phoenix effigy where a new exit has appeared. Dive off the cliff to get this achievement.
I can't take full credit for this, credit goes to Youtube user Martinoz. Here is his video (skip to about 1:30 to get to the EE steps, starting with Blake's outfit):