The purpose of this build is to do a lot of damage while being hard to kill, with this build you heal a lot and have a lot of health, you also have a lot of AD, do a lot of damage, and you do even more damage when at low health.
I actually don't know how to play Irelia, so if anybody who does tests the build, i would apreciate some feedback.

Conqueror is essential for the purpose of the build, it will give a lot of healing and damage.
Alacrity is very useful since the build lacks attack speed, something that irelia needs.
Last Stand will help survive when you're about to die, it also synergizes with Goredrinker's passive.
Ravenous Hunter will give more healing.
If you want more healing you can change Alacrity for Bloodline, it works quite well.

Goredrinker is the main focus of the build, the point is to heal, do damage, and being hard to kill. Goredrinker gives a lot of survivability, speacially when at low health and in crowded teamfights. Basically, Goredrinker fulfills the purpose of healing and being hard to kill.

TLDR: Healing go brrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr.

Muramana is your primary source of damage, when fully stacked, it gives a lot of AD (About 87 at lvl 18), and it also gives on-hit physical damage and damage to your abilities. The on-hit and ability damage are really usefull for your Q. It also helps with mana problems. But since these bonuses only come when fully stacked, you'll want to build it when you already have tear stacked.

TLDR: A lot of damage, build when tear is stacked.

BotRK gives a lot of on-hit damage, and also gives life-steal. Since the life steal is very good early, you might want to buy vamp scepter first back.

Titanic Hydra gives a lot of AD (about 48 at level 18 with goredrinker) and HP, it also gives on-hit physical damage scaling with your health, it makes you more tanky while still giving a lot of damage.

Death's Dance gives a lot of survivability, it gives 45 armor, and it's passive helps survive a lot. It adds a lot of survivability to the build.

Spirit Visage gives MR and HP, but the most important part is it's passive, wich amplifies all healing on you by 25%, wich is really good for Goredrinker, Conqueror and BotRK. This is an alternative for Death's Dance.