Welcome to my Irilea build.
I have mixed together my own build with help from sevral other builds and my own excperience.
So, this is how i play Irelia.

Before posting a comment about the item build, read this;
Pick only ome pair of the boots, depending on the team you are facing.
If there are more 3 or more stun characters in the other team, you want Mercury's Treads
If not, you can choose between Berserker's Greaves or Boots Of Swiftness.


Pros
Great Damage
Deals Truedamage on activate spell
The secound you get sheen, you will dominate
Good Survivibility
Passive health regen/hit
Lower time CC'ed
Slower/Stunner
Good Pusher

Cons
Poor mana pool early/midgame if not played correctly
No good escape ability (except slow/stunn, but that rairly works as an escapetool)
Having a hard time laning against ranged characters

The runes can be diffrent from what i have of course.
But to successfully be able to deal alot of damage you need to stay alive, yes?
so i go with Greater Seal Of Resilience for armor
and Greater Glyph Of Warding for Magic ressist.
That will give you some extra survivability.

Filling your rune page with DMG works aswell, but in my opinion ARP works better than AD.
With that said; i go with Greater mark of desolation and Greater Quintessence Of Desolation to get that extra Armor Penitration for some extra early lane domination.
Not only do they work as damage early game, but they also help you to kill the tank you are facing on your lane much easier.

Other runes to consider is Health or Ad runes.
Some people even prefer magic penitration for the ultimate and stun/slow ability.
So try yourself and change these till you find a matching runeset for your gamestyle.

For masteries i go with Improved Exhaust for escape or chase ability.
If you are chased by an angry feeded akali you wont get far without exhaust. Even if you flash she will be able to dash onto you again.
Not if you flash through a wall tho***

Im not saying that exhaust will help you survive an angry fed Akali, but your chanse of surviving will be higher for sure***

I also specc 3 points in perseverence the utility tree to gain a little more HP and MANA reg.

I go with 21/6/3 specc.

Go for the item order at top!
This is just a description.

So, i start out with Dorian's Shield for some earlygame survivability; health, HPreg and armor.
Then i rush to Boots Of Speed. You want to be fast so that you can lasthit minions and then run back to avoin unnessesary damage taken.
As soon as you got Boots Of Speed you want to save for Trinity Force; Starting with Sheen for an extra damageboost everytime you (Q) up to you enemy.
Then continue with Phage for a slowing bonus aswell as a DMG and HP boost.

Midgame is closing in. You want more lifesteal to stack with you'r (W) spell.
Buy Vampiric Scepter and then turn it into Bloodthirster.
Note; if you can afford BF sword before you buy Vampiric Scepter, Buy the BF sword first for massive DMG boost.


Now you got your basic damage and lifesteal items.
By this time in the game you might take alot of damage sence you are a Melee character.
Now you need to choose.
If you are facing a team with alot of Magic damage, you will want to build a Banshee's Veil. For magic ress, hp and the imune to one spell boubble.
If you are facing a lot of physical damage, well, build a Randuin's Omen.

After that you can fill in the missing spots as you want.
I tend to choose, Those few times ive come this far in the build, Infinity Edge; for a nice damage boost and high crits.
Or a Frozen Mallet for a little more HP.

So, farmin with Irelia is easy.
Some say that her (Q) Spell is great for farming, and it is. If you got the Golem buff.
If not you will probably just waste mana for what? 100-200g? Not worth it.

Hit minions with melee. Its the best way. And ofc, steal the big minion with (Q) from your teamate :P
When you get to mid/late game and run to clear a lane, use your ulti (R). Its fast and not as mana expencive. The CD is very short anyway.

Using your ulti to farm hordes of minions is less mana consuming that clearing with your (Q) spell. If you use Ulti, it will cost like 200 mana, but if you use Bladesurge it will cost alot more than that, because you need to spam it so many times.***


Constantly getting Golem Buff and Lizzard Buff will help you in sevral ways.
You get gold for killing them ---> Better Equipment.
You get buffs to help you fight ---> Longer time in lane,more money,more XP, more kills, ETC. = Better Equipment