Extreme mode has enemies doing 120% of the damage (vs. 25% on Low) that your weapons deal. When you do each map for the first time, play it on Medium. This will let you learn the layout of the level.
Once you've completed it on Medium, start running the map on High. Get to learn the enemy start/spawn points, and try to anticipate them. Make sure you bring the right weapons for each fight. On High, you take 85% damage, so you'll get a good handle on avoiding damage while not being punished too severely. Once you can do two back-to-back runs of the level on High without dying, you're ready for Extreme.
You will die a LOT in Extreme, especially in the beginnings of the maps, but you must persevere. Use headphones if you have them; hearing enemy footsteps is essential to getting the first shot when they're coming around the corner. Beware enemies toting the MP-50 or Type-7. Without armor, they can one-shot you! Replicas with the MOD-3 can one-shot you no matter how much health and armor you have. Try to keep them before they can get a shot off.
Don't be afraid to backtrack to pick up anything, but be mindful of one-way doors and gates. Armor and health boosters are best saved for when you have to move to the next area, ensuring you can soak some damage if necessary.
My Extreme map runs are listed below. These were recorded in 720p, so you should be able to see enemies in the distance easier. I try to show where the enemies spawn/enter from.
You may wonder why I opt to use melee a lot here. Weapon-based melee attacks do 150 damage (bare fists do 100), and only a few Replica types and Elites have more than that. Melee attacks do nothing against Heavies and REV6/8s. The special melee attacks (bicycle kick, slide tackle) do 1000 damage; they are an instant kill against any unarmored enemy. Making it connect is the problem.
There's also a quirk in the Replica AI that causes them to do a deer-in-headlights act for a few seconds after spotting you. This is very noticeable in Map 2. Shooting them in this state makes them return fire immediately, almost certainly damaging you. Running up for a melee kill is a safe strategy if you can close the distance fast enough.
Map 1 - Distribution
This is an extension of the end of Interval 3. Getting in position here early on is critical. Turtling up in the office is suicide.
In the third area, the Hind will not come until you take out the Replica with his back to you. Use this opportunity to stock up on grenades, armor, and Slow-mo.
In the third corridor, the REV6 will not activate until the door opens. If you stay out of the fourth area, the five Replicas will not spawn. This means you only have to worry about the REV6 and Replicas separately. The HVP does serious damage here, as it largely ignores its armor.
Map 2 - Rooftop
This is a remake of the beginning of Interval 4. Get ahold of the MOD-3 and use it against the two Replica below, then save the rest of the ammo for the Heavy at the end. If you throw your grenades against the walls in the same spot I did, you're all but guaranteed 2-4 kills from each one.
The Heavy in the elevator is packing a Type-7. With full health and armor, you can withstand ONE hit from it. The MOD-3 can kill him quickly, keeping him staggered the whole time. Once he's down, an elevator with three Replicas comes up. The door will open when it reaches 9.
Map 3 - Construction
Your main problem here is keeping the Assassins from getting the jump on you. Take them out one at a time, and make generous use of Slow-mo in doing so. Do not attempt to melee them, as the melee hit detection on Assassins is terrible.
Another thing to worry about (especially with the Type-7) is that there is an auto-aim function built into the game. If you fire a shot at an enemy completely in your reticle, the shot will home in on the center mass. This can result in what should have been a headshot completely missing, hitting a piece of cover directly in front of the enemy. Try to make sure the center mass is exposed when you take your shot.
Map 4 - Vault
Is is a kind of reversal of the escape sequence from the campaign. The MP-50 and MOD-3-toting Elites will be the end of you here. The two mines I drop catch an HVP and MP-50 Elite each. The other is unreachable to me before they arrive.
The HVP does 35 damage per shot, and the pistols do 40. The Nightmares have 35 health, making those two perfect weapons for fighting them. Grenades also work for getting multikills.
When the door slams in your face and three Elites enter, make sure you do not attack them until the first one comes into view. Otherwise, you can trigger a bug where the third Elite does not spawn, and you'll be stuck in that room.
There is a strange bug on how the MOD-3 Elite's door opens. The grenade I threw just stopped in midair until the door opened. Slow-mo activates automatically for a second, which can really throw you off guard.
It takes 12 HVP hits to destroy each appearance of Alma. Don't use the MP-50; you'll just liquefy yourself.
In the final fight, another two MOD-3-toting Elites come out. They are the first and last Replicas to appear after you clear out the Vault entrance. The prox mine I throw up there gets the first one. Then I just chuck grenades like mad and fall back when I run out. It is very important that you do not take your eyes off that catwalk. The Elites love to come out and waste you when you're looking away for even a split second. Don't try to camp under the catwalk. Their AI is wise to that, and they'll just stay in their spawn rooms until you come out.