This chamber is essentially a rather large elevator shaft with a number of shoot-able wall points all the way up to the top. The main trick here is to use enough momentum to get you up each floor.
As you approach the edge of the elevator shaft, zoom up and to the left, you'll want to place this portal just above the second walkway you can see. Now set the entrance portal on the wall just over to your right.
You'll land on the walkway, so once you're halfway around it, once again zoom up and to the left, there's a wall panel just under the bottom of the elevator that can be shot (see screenshot above for exact location).
Once you’re at the entrance to the elevator, drop a portal in the centre of it and then peer over the walkway to your right. Set the entrance portal down here, fall through it and you’ll sail up through the lift. Try and land on the nearby angled ledge (that has a light patch on it).
Set your exit portal on the white patch on the ledge, and peer back over towards the portal that should still be in the floor of the elevator. Fall down and into the portal and as you're ascending look out for the angled wall panel directly opposite the exit. Set your exit portal here.
Once again drop through the portal in the elevator and you'll come out of the portal on the angled wall and straight into the exit door!