Tough one, at least per my standards. So far only 1 unlocks I hope I can join him soon... :))
I would seriosly recommend the first playthrough on Easy, to get all other achivements there, focusing only on survival here.
The major difference vs the lower difficulty levels, is that if your guys are downed, even if you revive them, they will not join the fight until the next checkpoint (which is the end of the current fight). This makes most of the sections nearly impossible, as you'll have no access to their special abilities either, which makes killing large enemies a HUGE pain.
The main focus therefore, is to keep them alive. Progress very, very slowly, always keep them behind you if possible, as they can revive you anytime, while you'd lose them if they die. They tend to wander ahead relentlessly, so pause frequently and order them to stay put and only advance if you are 100% sure that they will be able to survive. Always wait for abilities to recharge if you can afford to stay in cover.
Most useful abilities for me were Lift, Critical Strike, Fire Support, Mind Control, Mine and Rocket Launcher. Cloak can also be useful for hiding them and send them to safe locations. Always deploy your blob whenever you can, its good to annoy them. Same for the drone, but here its a matter of taste: I went for healing only to find out that the healing rate is way too slow to make up for the incoming damage, therefore I think I should have chose attacking drone. If you know that a move will trigger a fight, place several mines (time consuming to wait for the ability to recharge) and try to lure them into the trap.
For weapons, always trade up for the new ones, and try to have once with quick fire rate and one with explosive effect.
Regarding smaller enemies, always kill the flying drones first as they shoot above the covers and once they fly close, you are toast. The healing ones keep on healing even each other, so its important to focus your entire team on a single target all the time. One Critical shot is enough to finish the first one once the encounters start. Next one should be Minion Commander, as they keep on releasing more Drones. After that, focus on Psionic Commanders as they Lift your guys, making them dead very fast. Then its Phantoms, Scouts, Snipers and Shield commanders, they can be equally tough depending on the battle scene setup. All the others are the last to finish.
And now onto the large ones. Battleships are terrible, unless you can find some cover under a roof. In that case you can hide there, finish all enemies in the distance and then even kill the plane with a pistol if you have enough ammo. If no cover, then unleash all you have on it immediately or your dead. Blasters and plasma guns work the best. All this is the same for the Titan as well, 3 Blaster shots then finish with your other gun.
Sectopod is another beast. Aim for the cockpit and once the cover is broken, go for headshots on the pilot. Other than that, just try to hide behind him :)
Gun turretts: At least they dont move, but can mow down you and your team in a split of a second. Sticky grenades are the most useful against them as you can easily land the on the top of them.
Mutons / Elite Mutons are the worst ones. At the beginning I dont have any special strategy other than try to hide and pop-out / shoot / hide back while keeping your team hidden. Later with developed agents, I make sure to have all abilities charged before they spawn, then lift them, place a mine underneath, a rocket launcher to the front of him, call a Fire Strike and a Critical shot on him, and shoot with all guns. This tends to kill them in 3 seconds, the problem is if you cant have abilities charged up before the spawn.
I hope I didnt miss anything, good luck to all for this challenge.