As a quick note, these guides are being written from the perspective of a player who is usually playing a Druid tank (and occasionally healer). I'll try to point out if I know of alternate strategies for different groups or tanks, but I certainly don't know them all. If there's a strategy you've successfully managed that I don't mention here, please drop a note in the comments.
AHN'KAHET: THE OLD KINGDOM
This is one of the harder heroics overall, but the achievements here range from very easy to fairly difficult. As such, it's a good heroic to use as a diagnostic tool; the individual achievements here will allow you to figure out if you've got the tank quality, healing throughput, and DPS firepower to get through the meta-achievement. If you can manage all of the achievements here, you should be able to manage everything else.
A Bloodlust will help a lot on the final boss and, depending on your group composition, may also help on the first.
Respect Your Elders
As you're probably aware by now, the Ahn'kahar Guardians which spawn periodically during this encounter render the boss and his titchy little adds immune to all damage. Ordinarily you handle this by killing the Guardian (they have about 35K health) and then going back to DPSing the boss and AoEing the adds, but here you're going to have to come up with a way of keeping the Guardians occupied elsewhere. The immunity effect is canceled once you get them out of range, which I believe is roughly 40-50 yards. You know the somewhat elongated room in which you'll find Nadox? Its length is a rough guide to the distance you need to maintain between Nadox and the Guardians in order to keep the latter from granting the former total immunity.
Off-tank (DPS DK, Warrior, or Paladin is fine)
Off-healer if needed (hybrid DPS is fine)
If your tank is very well-geared, you can usually pull this off without an offhealer. I've also had a group successfully use DPS Shamans as Guardian tanks here, but any sufficiently hardy class that can heal itself could probably manage it. Most strategies suggest having your off-tank pick up Nadox and your real tank pick up the Guardians because Nadox's melee is very weak but damage from the Guardians really starts adding up.
Now, I used to do this all in Nadox's room, but commenters in the Wowhead thread on this achievement noted a much better way to do it that I'll be using in the future. As you clear to the boss, take the path to your left, and clear all of the trash you find in the hallway immediately preceding Nadox's room. Have your off-tank aggro him, pull him out of the room down the stairs to the left, and tank him at the farthest end of the hallway you just cleared (if this doesn't make any sense, the commenter Caylie on the Wowhead thread provides pictures). Your DPS and healer stay here as well and just nuke down the boss. If you're situated correctly, Nadox will never go immune.
Your real tank will sit at the stairs and wait for Guardians to hatch. Because they'll make a beeline for your healer, they'll head down the stairs toward Nadox and you can pick them up very easily and keep them tanked here. You may need an off-healer to toss you a few heals because you will be well out of range of the real healer. However, if you're sitting on some decent gear, blowing a few cooldowns and a health pot is usually sufficient if your DPS is competent and Nadox dies quickly.
The Party's Over
Fairly simple. The most obvious way to do this is just to have someone drop group and leave the instance, but apparently you can also have someone drop group, wait for the rest of the group to start DPSing the boss, and then participate as you normally would. The person who drops group doesn't get the achievement but can still help on the kill. Weird.
If your tank and healer are well-geared, this isn't tough. Jedoga's damage will increase substantially when her "sacrifice" isn't nuked down, but it's really nothing that a few cooldowns -- or, failing that, a lot of healing -- can't handle. If your tank is still having difficulty living through this, try to have an off-healer in the group who can help out a bit. If that still doesn't work, you can attempt to rotate taunts on Jedoga across her platform -- but with diminishing returns on taunts in 3.1, I think it's safe to say that you should treat this as a tank/healer gear check more than anything else.
Volazj's Quick Demise
This achievement is significantly easier with a Protection Warrior tank, as they can Spell Reflect Volazj's Mind Flay and cause a ton of damage to him, but it's still doable with any class of tank provided they don't threat-cap their DPS.
This achievement becomes fairly straightforward once you realize that the real boss isn't Volazj but rather your Insanity adds. Volazj himself will actually die quickly to competent DPS; it's the adds for whom you need to be blowing cooldowns and potion timers. With a Bloodlust/Heroism and good DPS, you will probably find yourself triggering both Insanities at the same time. If your group has the capacity to deal with these adds within 30-40 seconds, congratulations -- you can do this achievement with time to spare. If you can't, time to start tinkering with group composition or some of the more punk ways of managing this achievement. The good news is that there are a lot of ways to manage this (the Wowhead comment thread lists most of them), from Warlock summons to Army of the Dead to Divine Intervention to having your healer suicide straight off the pull or just come as DPS. The bad news is that none of these will compensate if you just can't pump out the damage you need.
My groups tend to go for the straight burn method, using Bloodlust at the start of the fight and triggering both Insanities at the same time. Using Army of the Dead going into the Insanity phase is extremely helpful if you have a Death Knight, as is stacking your group on top of each other to minimize time spent getting to ranged adds for your melee DPS/tank. Blow all of your serious survival and DPS cooldowns on the adds and get back to killing Volazj posthaste. If you manage this, you should have plenty of time to kill Volazj before the 2-minute mark. The fastest I've managed it is 52 seconds; the slowest has been 1 minute and 27 seconds, so I tend to favor this over more gimmicky techniques because there is room for error or slower/less bursty DPS.