At the end of the first path there’s a light wall in front of you and through a wire fence to your right, use your portals to step through to the other side of the fence, now look up to your left and replace the farthest away portal with one on the horizontal roof.
As soon as you reach the 'DO NOT ENTER' sign, turn your view up high to your left, there’s a walkway up here with a white wall panel, fire a portal up here and then another one at the wall just around the corner to your right.
The end of the walkway overlooks a small white wall below you to your right, get a portal down here and then the other one on the wall directly to your left. Hang a right and drop down the ledge here and then pull the lever to open the metal door that leads into the open area.
Replace the room 2 portal on the wall behind the button meaning you should now be able to immediately traverse between both rooms. Hit one override button and then turn right around and hit the next one to - slowly - open up the, rather large, vault door up ahead. Enter it to start the next part of Chapter 6.