Now some words about the instances in the Storm Peaks:
Halls of Stone - the encounter Tribunal of Ages can bug and leave you without being able to loot the chest that contains the loot. Personally I have found that it will reliably bug when I take my Paladins and only spam Consecration, so be sure to use direct spells/attacks in addition when on this class. For other classes, I have yet to be able to bug the encounter, so it should be somewhat safe to just do the usual spammy attacks.
Ulduar is a raid that have a conditional hardmode that gives you the chance to be rewarded additional loot from respective bosses. Each boss with hard mode available requires you to do something specific to that boss to enable their respective hard mode for that boss.
Also Ulduar was previously available as both N10 and N25, but is now merged together to N10 for timewalking compatibility, and all old N25 loot is available among with the old N10 loot in the current N10 version. This doesn’t matter in regards to hardmode anyway, just FYI.
There is an ongoing rumoured issue that some bosses doesn’t drop some items from their hard mode loot table. Blizzard have yet to respond to this, and I have yet to personally confirm this issue.
That said, here is an overview of the bosses and how to enable their individual hard mode:
Flame Leviathan - speak with the titan guardian at the start of the raid and opt in for activating the Secondary Defence System. Once the NPCs are reacting to your action, you can mount up (yes, DO NOT enter any vehicles as the vehicles scales you down to raid level!) and move inwards to the gate where the boss comes through.
The hardmode is here determined by how many defensive towers are left active, so just ignore them completely.
Razorscale - does not have a hardmode.
Ignis the Furnase Master - does not have a hardmode.
XT-002 Deconstructor - this is the most difficult ones for highlevel characters and [Whole-Body Shrinka] is recommended, and remove any gear and talents that procs passive damage! Engage XT-002 with a non-covenant spell/attack, and get them down to about 70% health. Then they will become unattackable and spawn a hearth. Kill the hearth. Doing so will give XT-002 100% health again, and the boss can now be killed.
Some Covenants spells and effects DOES NOT scale down with [Whole-Body Shrinka], so better be safe than sorry.
The Assembly of Iron - consists of three mobs in a pack. Again, make sure not to have any passive damage procs or anything like that, and only use single target spells without AoE. Steelbreaker MUST be the last mob alive. It doesn’t matter in which order you kill the two other mobs.
Algalon the Observer - is a hardmode boss by default, no actions needed. Just DO NOT nuke him whenever he becomes attackable as he will then instantly nuke you to zero health. Engage him with a dot or an auto-attack or something else that doesn’t do too much damage, and then blast him down if he already haven’t withdrawn from combat (aka “dead”, but technically he does not die.)
Kologarn - does not have a hardmode.
Auriaya - does not have a hardmode.
Hodir - simply nuke him. The hardmode challenge is to kill him before he destroys the chest on the right side of the room.
Thorim - also simple, just nuke him. It can indeed be annoying to get to him as you need to kill everything in the arena at once before you can interact with the handle that opens up the way upstairs to reach him. The hardmode challenge here is a combat event timer which is no problem for highlevel characters. Just remember that when upstairs and fronting the hall with Thorim at the end, walk around the big circles on the floor and dodge the stun hammer being thrown at you.
Freya - another easy one. Don’t clear the area, we need the big tree mobs alive. Head straight to Freya and nuke her. That’s it. Afterwards perhaps even take the opportunity and pick some flowers if you are a Herbalist as the area is full of herb nodes.
Mimiron - work your way in and enjoy the tram ride. When entering his workshop, DO NOT attack him. There is a huge red “Do not click”-button on the wall behind Mimiron. Walk around the room and click that button. doing so will automatically start the combat event. Now you can just nuke through the phases as normal.
General Vezax - is almost as sensitive as XT-002, but this time you don’t need to do any damage to the boss. Make sure you have no gear, trinkets, talents or anything at all that does passive damage. Then you can go ahead and bodypull General Vezax, and you should position yourself towards a corner in the wall behind him so you don’t get kicked around the room. Hold your position and go AFK for some minutes until the boss spawns a Saronite Animus - which only happens if you ignore the spawning Saronite Vapors. Animus reduces General Vezax’s taken damage by 99%, and you can now kill the spawn and then the boss himself.
Yogg-Saron - whenever you freed the titan keepers from the influence of Yogg Saron, they will stand around the central room and offer a buff as their way of helping you take down this last boss. Normally this is promptly ignored as highlevel characters doesn’t need the buff to down Yogg-Saron.
This is relevant to the hardmode, as it’s legacy difficulty and following chance for hardmode loot scales with the amount of titan keepers that helps you with their buffs during the encounter. Since we highlevel characters don’t have to care about getting helped by them, we get the hardest hardmode and best chances for hardmode loot automatically without having to do anything.
If someone think I have misunderstood how to enable hardmode on some bosses, please let me know. I am after all focused on Ulduar in my weekly raid clear at the moment. 
Edit: added additional hardmode details for Flame Leviathan and General Vezax.