To access the keys and the exit on Doom II level Tenements:

(Note: For a YouTube video covering this topic click here.)

Before you can exit you'll need to pick up all three keys.
If you have them already, skip down to "THE EXIT", below.

From the level's starting point, walk north up the walkway
in the green slime.  In the room ahead, turn left and go
through the open doorway to the west.
Continue west down the narrowing zig-zag staircase, and
pick up the red key at the bottom.

Go back up the stairs and back through the doorway at the
top.  (If the door is closed, you can either wait for it to
open again, or you can jump south off the stairs and step
on the protruding corner of the teleporter pad in the
southwest end of the red pool, which will teleport you back
to the other side of the door at the top of the stairs.)

Once you have the red key, and are back through the doorway
at the top of the zig-zag stairs, look for the twin flights
of steps nearby, and go up the west-side steps.
Go through the red-keyed door at the top, and Activate the
switch you'll find in the southeast corner of the room.

Go back down the steps, and continue ahead, up the other
flight of steps to the east.
Look for the switch on the wall in the southeast corner of
the room, and Activate it.
An elevator under your feet will drop down, placing you at
the northwest corner of another room.

Go south, down the narrow hallway, and around the corner to
the right, where you'll find a switch.  Activate it and a
lift will drop down to your right.  Get on and ride up.

Step forward into a small room, look to the left for a
switch on the south wall and Activate it.
Go to the northwest corner of the room where, just behind a
narrow partition, you'll find another switch mounted on a
wall panel.  Activate it and the panel opens.
Step through to drop down to the slime below.  Run forward
to get to the platform ahead.
Turn left and proceed north, through the open door (which
was previously closed and locked).

You'll be at the south edge of a wide, square slime pool.
Run through the open door on the east side of the pool, and
head up the stairs.
Continue east to the far end of the passage, and you'll see
steps to your left and right.  Go up the left (north) steps
and look for a torch in the northeast corner.
Activate this torch like a switch, and a wall panel at the
bottom of the steps behind you will open momentarily.  Go
to it and step through before it closes.

Go south, up to a small chamber, and Activate the switch
you'll find there.  Step west through the opening in the
wall, and you'll drop down to a platform below.

Continue straight across the narrow bridge that leads west
to another platform.  Drop into the crevasse dividing this
platform, and quickly Activate the switch at its north end.
The floor across the bottom of the cravasse will begin to
rise.  When you get to the top, head east back across the
bridge, and turn left.

Go north down the steps, up the other steps and turn left
again.  You'll see a new bridge leading west to where you
can get the blue key.

Once you have the blue key, exit this area by going back
out to the large square slime pool located in the middle of
the map.

Go through the blue-keyed door on the western edge of the
slime pool.  Once inside, go left to the south side of the
room, where you'll find a staircase.
Start up the steps, and when you get to the top, you'll
cross an invisible tripwire that temporarily opens a small
chamber near the bottom of the steps.  Run down to get into
the chamber and collect the Soul Sphere there.

Exit the chamber and go back up those same steps.  At the
top you'll now find a switch.  Activate it, then go back
down the steps and head to the north side of the room, and
up the other staircase.
At the top is a newly revealed chamber.  Step inside and
look to the left for an opening in the wall.  Walk in, and
follow the path around to the left, heading west, then
south, until you come to a chamber.
Look left/east and you should see an opening that looks out
and down over a smaller slime pool in the large room you
were in a moment ago.

From here you should be able to see the yellow key resting
on top of a raised lift on the east side of the slime pool.

Jump down into the slime and Run east to get on the lift.
Ride it up and grab the yellow key.  Step east off the
platform to drop down to the floor of the room, then head
east back out to the large central slime pool.

THE EXIT
--------
Once you have all three keys, look on your Automap.  In the
center is a large square slime pool.

Go to the north edge of the slime pool to where you'll find
a booth that can be opened with the yellow key, revealing a
switch.  Activate it to open a path leading north from the
slime pool, and head that way.

Continue north through this large room to a raised walkway
that splits to curve up and around to the left and right.
Take the left path.  When you get to the end, back up a
little and then Run off the edge to land back in the raised
western half of the room.

Look for a door on your right.  Go through it and follow
the stairs clockwise around the room until you reach a
switch, and Activate it.  Step back out the door you came
in, and look across the room in front of you.  You'll see
that another door is now open in the east-side wall.

Go through that door and look to your left.  A switch there
will drop a wide elevator-platform next to you.  Get on and
ride it all the way up.

You'll come to a room with some blue torches in it.  Walk
up to the torch in the southeast corner of the room.  There
should be a plasma rifle here too; grab it.

A panel will open in the north wall of this room.  This is
the entrance to one of three winding passages that will
eventually be accessible from this room.  Each passage is
made up of smaller, separated sections.  To open up a
section in one passage, you need to walk to the far end of
another passage.

Below is an approximation of the passages as they will
(eventually) appear on your Automap.  Follow the numbered
positions in the diagram to open up all the passages.  The
center passage leads to the exit.
               _
         _____|4|            ____
        |2 _____|     ______|    |_
        | |  ________|6 ____   8  _|
 _______| |_|  ________|    |____|
|             |            ___
|             |________   |  7|
|            1 ______ 3|  | |
|_____________|      | |__| |
                     |_____5|

Standing in spot 1 will cause a panel in the north wall to
open, allowing access to spot 2.  Standing in spot 2 will
open a panel in the east wall, allowing access to spot 3.
Standing in spot 3 will extend the north passage, allowing
access to spot 4...
Reaching each successive spot will open a new passage, or
extend an existing one.  Continue until all three passages
are open and extended as far as they'll go.

Once you've reached spot 7, the exit chamber (8) will open.

PC Doom II players can click here to access a demo showing
how to acquire all keys and reach the exit on this level.