We've gotten Archimonde down once on normal, and it was pretty tough. Here are the strategy adjustments we needed to make:
* For phase 1, we tried moving around the edge of the room (which seems to be the dominant strategy from videos) and a ping-pong strategy where you bounce between predefined points for each allure of flames. Ping-pong seemed to work better for us. We used two melee positions about 50 yards apart, and parallel ranged positions about 20 yards behind them. Doomfire targets kite the fire in a loop around the outside. Depending on raid DPS, there can be an awkward time after the second Allure where Archimonde holds onto his shadowfel burst until after he summons a doomfire (so the raid is spread out when the doomfire picks a fixate target), but it's not too hard to adjust to with practice.
* For phase 2, we kill off any remaining doomfire and have the ranged close into the melee, still ping-ponging between the two melee positions for each allure. Wrought/focused chaos targets move slightly out of the raid and point the ray sideways or away from the group, which is the normal strategy. This was never a terribly difficult phase for us; when we had problems, it was usually from people taking too many stacks of doomfire while moving for allure. Announcing allures several seconds in advance so people can move to the inner edge of the group fixed that.
* The beginning of phase 3 isn't hard. The first banish gets run to a point on the outside of the room and the first team goes in; we sent in five people (three DPS) for a 15-person raid. The boss is dragged to the center of the room, around 50 yards from the center of the tear. We appoint a ranged DPS who isn't on a banish team to stand near the first nether tear and snare/kill the adds which come out.
* The hard part starts when the boss hits 25% and the first rain of chaos happens. A non-obvious point of strategy is that you don't really have to dodge out of the rain of chaos impact points; it's better to stand still, use minor defensive and raid healing cooldowns, and maximize DPS. We tried several strategies for dealing with the infernals, but mostly settled on a free-for-all with maybe a couple of people trying to keep aggro on one infernal each to make sure they don't heal each other. They hit fairly hard on a DPS but it can be healed through. We generally wind up letting some nether tear adds get through while killing the first infernals. We hit heroism after the first set of infernals is dead.
* The second banish happens around the 25% mark, perhaps a bit before. We run it to a point along the edge next to the first nether tear, ideally chosen so it's not too long of a run. Getting the second banish team in was a bit of a pain point for us, so calling out this banish with plenty of warning is useful. It's nice if the boss doesn't get moved too much out of position so the raid doesn't lose too much DPS during this process. We appoint a second ranged DPS to stand next to the second nether tear and snare and kill the adds.
* When the third banish happens, we send in a suicide squad of just three people (tank, one healer, and one DPS). We run the third banish to a point in between the first two nether tears (making an equilateral triangle); again, it's important to announce this banish in advance so no one misses the target, as the boss heals by 5% if fewer than three people go in. This banish starts a clock; we have to win before the fourth banish or we have no tanks. Three nether tears is really hard to keep up with anyway. When the second rain of chaos happens, we ignore the infernals burn the boss (also ignoring nether tear adds). This spawn starts a second parallel clock; the infernals will wipe the raid after being alive for 35 seconds, so we have to win before then or very shortly after.
* Inside the twisting nether, with five-person groups, it seemed to generally be more reliable to commit to killing two small adds than to try to beat the second add. It depends a lot on the DPS composition, though.
With this plan, we win if nothing goes wrong in phase 3: nobody dies during the first infernals (or maybe one person if we res them quickly), nobody misses their banish, nobody dies inside the twisting nether except the suicide squad, and nobody dies too quickly during the second rain of fire. We had an unfortunate number of wipes around the 2% mark because of one too many things going wrong.
I think it's possible to send in a real team for the third banish and expect to kill the boss with four active nether tears. It's probably more manageable in a larger group; in a group of our size, you start running out of ranged DPS to assign to add management.