Go through the Emancipation Grill up ahead and go through the lobby area. You find another Elevator to Surface, and this time you can actually open the door. The elevator isn't there, though. But some of the walls are white, so you can use portals to make your way up there.
Look up the shaft and zoom in to see the white walls up there. Put a portal on the highest wall you can see. Now put another near you, and go through. Your objective is to get into the elevator, so if it's still above you, keep using white walls to go farther up the shaft.
The elevator itself has a white floor. Once you can reach the elevator, put a portal on the elevator floor. Now look around for a white floor somewhere below you. Put a portal of the other color on it and fall in. You'll come out through the top of the elevator. As you reach the top of your jump, move to the side so you land on the piece of wood sticking out nearby.
There are two white floors on the same level up here near you. Look at the portal in the elevator, then put a portal of the other color on the white floor next to the blank wall (in other words, not the white floor next to the wall with numbers and lights). Fall into the elevator portal.
As you go up, you will see an angled white wall. Keep in mind what color the elevator portal is, then put a portal of the other color on that angled wall. Drop back into the elevator portal. You'll be deposited on a ledge where there is an exit door. Go through.