To get the keys on Doom 64 level 13, Dark Citadel:

BLUE KEY
--------
From your starting position at the beginning of the level,
take a sharp left and head down the hallway.  Follow the
hallway until it makes the first turn to the right, then
look for a large open room on your left.  This room is full
of narrow pillars, running floor-to-ceiling.  In the center
of the room is a pillar which is slightly wider than the
others.  On the back of it is a switch; Activate it (this
opens a door that you'll need to go through in a minute).
Go back to your starting position; and now follow the main
hallway in the other direction, until you come to a door on
your right.  Step through the door and take a sharp right.
Follow the narrow passage up some shallow steps and around
to the left.  Keep going and you'll drop into a somewhat
wider passage with 3 long, horizontal windows to your left.

(Note: The second of the 3 long windows is open, and you'll
be able to see an Invulnerability Sphere on a pillar below.
You won't necessarily need this now, but take note of its
location, as you may want to come back for it.)

Near the far end of the passage with 3 windows is a plain
square room on your right.  Step into this room and the
floor drops down, granting access to a large, low-ceilinged
room with 14 thick pillars spread out within.
Step into this room and turn right.  Follow the right-hand
wall until you come to a small opening on your right.  Go
in and look for a switch.

When you activate this switch, the thick pillars back in
the large room open up, releasing a prolonged blizzard of
flying metal spikes, and the blue key will appear on the
floor in the center of the room.  After several seconds,
the blue key will disappear as the pillars close up again.

You can attempt to run out and grab the key before the
pillars close up again, and it is possible to do so without
taking damage, but that can be very difficult.  For this
reason, you may wish to go grab the Invulnerability Sphere
mentioned earlier.  If/when you do get it, immediately head
left from the pillar where you picked it up, and Run for
the lift you'll find several paces ahead of you.  Ride the
lift up, turn left and go back through the 3-window passage
and back into the room with the blue key.  If you're quick,
you'll have just enough time to activate the switch, grab
the blue key and get away before Invulnerability wears out.

RED KEY
-------
Start from the room where you pick up the blue key.  Look
for the lift that let you down into this room.  The lift
will appear as a textured surface in a large rectangular
indentation in the wall.  Walk up to it and activate it
like a switch, and it lowers.  Step on and ride it up.
Exit the lift/room and look for 3 long, horizontal windows.
Go to the middle window, and run through, dropping down
into a huge area with skylights and more pillars.  Go
forward, following the wall to your right counter-clockwise
around the room.  You'll eventually come to 3 blue vertical
bars on your right.  Activate them to open them, and then
activate the switch you'll see there.
Now find a doorway on the opposite side of the room, and
once there, look for a switch on the inner right side of
the doorway.  Activate this switch to lower a lift in front
of you.  Step on and ride up, then walk forward, past the
openings to your left and right, and back out into the main
hallway.  The main hallway runs around the middle of the
level in a square pattern.  Follow this hallway in either
direction, until you get to a darkened corner.  (This
corner will be diagonally opposite to the corner where the
map's original starting position is.)
A short "stalactite" hangs from the ceiling in this corner,
and on the back of it is a switch.  Activate it, and a
moment later the floor under you drops down.  Turn and look
for an opening on the right.
As you walk through this opening, look for a similar
doorway ahead of you on the far/opposite wall.  Go through
that opening and into the next room.  Search through the
bookshelf-maze in this room for a switch.  It'll be at the
back of a dead-end in the small maze.  Activate this switch
then leave the room.  As you come out of the passage
exiting that room, turn right, and follow the right-hand
wall, until you come to the second doorway on your right.
Find your way through this maze until you come to another
passage leading up to a glowing pillar in the middle of a
small room (this will be a room you haven't been in
before).  Head to the pillar and look for a passage on your
right.  Follow the passage to a switch off to your left.
Activating it will drop a lift down.  Get on and ride up to
where you'll find the red key on your left.

YELLOW KEY
----------
From the point where you pick up the red key, jump down
into the room ahead of you.
Now find a doorway a ways off, around the corner on your
right, and once there, look for a switch on the inner right
side of the doorway.  Activate this switch to lower a lift
in front of you.  Step on and ride up, then walk forward,
past the openings to your left and right, then across the
main hallway and up the shallow steps ahead of you to the
red door.  Open the red door to reveal a large courtyard.
In the middle of the courtyard is a small fire, and in the
fire rests the yellow key.