This boss is another council style fight where the hardest part is at the start of the fight when all the adds are alive. Fathom Lord Karathress begins with 3 adds, Sharkkis, Caribdis and Tidalvess. Each of these adds have unique and annoying abilities but the most annoying ones comes from Caribdis, as she is the healer of the group as well as the main CCer. Hence Caribdis is focused down first. Once Caribdis is down, the order to defeat the remaining two adds doesn't really matter, but it is recommended to kill the boss last.
The boss himself has the most annoying ability of all that is *apparently* only used against mana users, eg shadow priests. It's called Cataclysmic Bolt and it deals 50% of its target's max health as damage plus an additional 2 second stun. Fortunately this ability can be slightly mitigated with shadow resistance. But the fact that shadow priests are susceptible to this ability to begin with... already puts them at a severe disadvantage compared to DKs and Hunters. The way to cope with this ability during the fight however, is to simply heal through the damage.
Begin the fight by dotting up Sharkkis (or Tidalvess) followed by fade, and then send the shadowfiend onto Caribdis. The reason I used fade BEFORE sending out the shadowfiend is to simply get an extra GCD I can use on the t13 4p rotation. Even if fade is used before sending out the fiend, threat will still be dumped onto the fiend as long as it is sent out before the fade duration ends. The reason that Sharkkis was dotted up before focusing on Caribdis is simply to trick her into healing Sharkkis in case she doesnt drop fast enough.
Once Caribdis is down, her abilities will be transferred to the boss so there are still plenty of CCs to watch out for. But at this point it's just a matter of surviving against the incoming damage and let dots do the work. Both Dispersion and PW:S glyphs helps when it comes to surviving. Additionally, Desperate Prayer is also a nice talent to pick up for emergency healing. Since CCs still get dished out quite often it is recommended to prioritize healing at below 50% health.
Eventually the remaining adds will go down and the fight becomes a lot easier at this point despite the boss himself now possess the abilities of all three adds. At this point just keep health levels high and hopefully shadowfiend will be up again for a final burst to finish off the boss.