In a nutshell, this boss is a DPS check.

The boss itself does slightly lower damage compared to most other Naxx bosses but it places a 15 second debuff which reduces healing taken by 10% and he places one stack of this debuff every 10 seconds. This means that as the fight goes on, the ability to self heal diminishes. The way to counter this is to not use direct heals when this debuff is active, and instead use PW:S which is not affected at all. Obviously at one point this debuff must be somehow reset in order to self heal and the way to do this is to send out a shadowfiend at 7 or 8 stacks (or more) of this debuff and fade. This enables the shadowfiend to tank the boss and the priest to drop all stacks, and hence enables to self heal. This strategy should be repeated of course. However Fade no longer seems to dump aggro after the third Shadowfiend summon (not sure why) so the only way to reset debuffs after that is to simply kite the boss, and hopefully the boss turns around and eats a zombie instead of applying an additional stack of the debuff. This delays the next application of the debuff and hopefully this small time gap is enough to reset all the stacks.

Gluth also occasionally gains an enrage buff for 8 seconds that increases his haste and damage by 25% each. There isnt much to be done about those.

As mentioned earlier, the boss eats zombies throughout the encounter. Normally these zombies are kited by players so they do not come close to the boss to be eaten. However since this is about soloing, the prevention of zombies being eaten is impossible. Gluth will eat a zombie every 10 seconds or so and this heals him for 5% of his max health.

Hence the encounter is all about overpowering his healing (hence DPS check) as well as using PW:S and other defensive CDs such as dispersion effectively. Aside from using PW:S on CD, it is also more effective to use dispersion when kiting or when Gluth gains that enrage buff. It is also worth noting that Gluth also has a hard enrage timer of 8 minutes which further stresses the fight as a DPS check.