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.
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).