To get to the secret exit in level E4M2, Perfect Hatred
(and how to access level E4M9, Fear):

If you already have the keys, skip down to THE SECRET EXIT,
below.

First, you need to get to the lava pool in the large room
just to the west of the level's starting point:
When you start the game, Run up and off of the left-hand
stairway, to land on the west-side platform ahead of you to
the north.  (In PC Doom games, "Run" is typically enabled
by holding down the Shift key.)
To your right is a pillar with a switch on it.  Activate it
to open a door high up in the west wall, behind you.
Run west off the platform you're on to jump through that
new doorway in the wall.
Just inside and below that doorway is a small platform,
down in the lava pool that covers the floor of this room.
Drop down to that platform.
Just to the north of that platform is a small triangular
area of the corrosive floor.  (On Skill levels 1, 2 and 3,
there will be a radiation suit here; on Skill 4 or 5, it
will be missing.)
If you step into this triangular spot, a narrow section in
the middle of the north wall of this room will rise up,
revealing a stairway that leads up and north.
Step into that triangle, and quickly step back onto the
platform.
Next to you, at the east end of the platform, is a switch
on the wall, located just under the raised doorway you came
through a moment ago.
If you Activate this switch, a walkway will begin to rise
out of the floor.  This walkway will extend all the way
around the outer edge of the room.
Activate the switch, and, if you move very quickly, you can
Run south to get on the walkway as it rises.
(If you didn't get on the walkway in time, leave this room
by going north up the stairs, and Run for the red teleport
pad you'll see on the floor ahead of you to the north.  It
will take you back to the level's starting point.)

Once you're on the walkway, follow it clockwise around the
room.
You'll come to a small gap in the walkway.  Run east to
cross that gap to a widened area ahead where you'll find
two green torches.
As you step into this area, a part of the back wall will
open up, revealing the yellow skull key on the floor of a
small chamber.
Pick it up, and go back to that gap in the walkway.  Drop
down into the gap and turn north and go up the stairs
ahead.
You'll see a red teleport pad on the floor, several paces
ahead of you to the north.  Step on, and you'll be sent
back to the level's starting point.

Run up and off of the right-hand stairway, to land on the
east-side platform ahead of you to the north.
To your left is a pillar with a switch on it.  Activate it
to open up another platform ahead of you to the north.
Run to get across the gap to that platform.
You'll see that a part of this platform narrows and extends
off towards the northeast.
Run northeast to land on another narrow projection which
continues northeast to a large, irregular room.
Enter this room, and continue southeast to drop down off
the platform you're on.
Proceed south around the various obstacles and look for a
stairway in the middle of the south wall.
Follow the stairs up and around, and you'll see a teleport
pad ahead of you.
To the left of the teleporter is a small chamber to the
south.  The blue skull key is here on the floor.
The teleporter will return you to the level's starting
point.

THE SECRET EXIT
===============
Once you have both keys, go to the large main room in the
center of the map, and look for the tall, winding stairway
at the north end of the room.
Access to the secret exit is through a door that is located
in a corner of this room, to the west of the stairway, down
in the lava just below and to the left of the window in the
west wall.  Here's how to open that door:
Raise the blue-keyed barrier column that blocks access to
the stairs, and walk up to the top of the stairway.
At the top of the stairs, turn to face south, looking back
down on the main room.
Step backwards a few paces to cross a floor trigger on the
wide platform behind you to the north.  You will see that
a column in the center of the room rises up, revealing a
glowing teleport pad.
Turn right to face west, and you will see the teleporter's
destination point, a small rectangular alcove which is
recessed behind a low horizontal barrier.
Go back down into the large main room and step on that
newly revealed teleporter.  You'll reappear in the alcove,
facing the barrier.
Step up to the barrier and Activate it like a switch.  This
both lowers the barrier, and temporarily opens the doorway
that grants access to the secret-exit switch.
When the barrier drops, quickly step past it, turn right,
and Run southeast, dropping down into the lava, and try to
reach that doorway before it closes again.  Remember, the
doorway is down in the lava in a corner of the main room,
across from the winding stairway, and just south of the
window on the west wall of the room.
(If you don't get to the door in time, Run counterclockwise
around the perimeter of the room, until you come to an open
doorway in the southeast corner.  Go though and follow the
stairs up and around, until you get back to the level's
starting point; Then return to the central teleporter and
try again.)

To access a downloadable demo which shows how to get to the
secret exit on this level, click here.