Hello and welcome to my newest guide, I am a Veteran League of Legends player , I've been playing since season 1 and I am currently a Warwick main.
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Also be sure to check out some of my other guides in the links below for more awesome solo Q shredding and rank ladder climbing.
Reworked Jungle Swain
Unstoppable Warwick
Tiger Godyr
Shadow Assassin Kayn
In-depth Kai'sa

Ionian Fervor Passive - Irelia gains 1 Ionian Fervor stack for every enemy champion hit by her abilities, or if she hits at least one non-champion, stacking up to 4 times. Ionian Fervor lasts for 6 seconds, with basic attacks against champions refreshing the duration.

IONIAN FERVOR: Irelia's basic attacks deal 2.5 - 16.5 (based on level) (+ 2% - 4% (based on level) AD) Ability power bonus magic damage per stack. At maximum stacks, Irelia gains 20% / 30% / 40% Attack speed icon bonus attack speed and deals an additional 20 - 132 (based on level) (+ 16% - 32% (based on level) AD) magic damage against shields.

Bladesurge Q - Bladesurge has both a passive and an active effect. PASSIVE: Enemy champions and monsters hit by Flawless Duet and Vanguard's Edge's initial barrage are marked for 6 seconds.

ACTIVE: Irelia dashes to the target enemy, dealing Attack damage physical damage, applying on-hit effects and healing herself. Bladesurge deals 60% increased damage to minions.
The cooldown on this ability is refreshed if the target hit was marked or it dies to this ability.

Defiant Dance W - FIRST CAST: Irelia aligns her blades around her, gaining 1 Ionian Fervor Ionian Fervor stack and channelling for a couple of seconds, during which she reduces incoming damage. Defiant Dance's channel cannot be interrupted.

SECOND CAST: Irelia swipes with her blades in the target direction, dealing physical damage to all enemies around her and in a line, increased based on channel time.
The SECOND CAST activates automatically toward the cursor's direction at the end of the duration.

Flawless Duet E - This is a long range line skillshot with a stun and a double activation is needed in order for the ability to activate.
Irelia sends a blade to the target location for up to 3.5 seconds. Flawless Duet may be recast to send another blade out, and does so automatically at Irelia's position if the first blade would expire, though not if she is affected by ability-inhibiting crowd control.

Once both blades have been placed, they fly toward each other after a brief delay, dealing magic damage and stunning all enemies struck for 0.75 seconds. Monsters receive 50% damage.

Vanguard's Edge R - Irelia launches a barrage of blades through a wide line in the target direction, exploding outwards upon hitting a champion, dealing magic damage to all enemies struck.

The blades form into a spade-shaped perimeter around the explosion for a few seconds, knocking aside all enemies. Enemies that pass the blades are dealt magic damage, and are disarmed while being slowed by 60% for 1.5 seconds. This ability is extremely useful in teamfights and should be carefully used.

A unit that is disarmed is unable to declare attacks.
It interrupts The Culling, but no other abilities it seems. The duration is reduced by tenacity, which stacks multiplicatively.
Resisted by Black Shield, Ragnarok, Riposte and Void Shift. Dashes that grant immunity to disables will ignore the effects of Disarm for the duration of the dash, but the target will be affected upon landing if the disable duration permits, which means that Warwick could still possibly be disarmed.
The disarmed effect can be removed by Cleanse, Quicksilver Sash, Battle Roar, Ragnarok , Remove Scurvy and Unbreakable Will.
We're going to be maxing Flawless Duet E first for the increased stun duration and then Bladesurge Q second for the increase in damage. Defiant Dance W should be maxed last as its damage isn't as high as Bladesurge and we're already going with a tank build so the extra damage reduction is not needed until the later phases of the game.
Down below we have the official Irelia Champion Spotlight done by Riot Games, showing the abilities and how they work.

Play Video

Now I'm going to explain everything about the runes in as much detail as I possibly can, pay attention as these are all important on Irelia.
The first rune build is the following:
PRECISION + RESOLVE and we get +9% attack speed and +15-135 Health at levels 1-18.
Press the Attack is our first choice, while possibly not as good as Conqueror in a 1 v 1 situation, this is probably the best if you're a team player looking to weaken enemy tanks, which you can easily do with a combination of abilities such as Flawless Duet + double Bladesurge due to the marking of Flawless Duet which enables a Bladesurge reset,so does Vanguard's Edge but that should be used for other situations such as denying the enemy carries from attacking.
Triumph this is a great rune to enable us to get our items faster as it gives us 20g per takedown and restores 12% of missing HP.
Legend: Alacrity is great for your attack speed as it adds more on takedown up to a cap, good for fighting.
Coup de Grace this rune adds a bunch of damage on low HP targets which enables Bladesurge resets as this ability also resets it's cooldown if it kills.
Bone Plating great anti burst rune which is needed as Defiant Dance has a pretty high cooldown, this is going to keep us alive most of the time and deny massive damage.
Chrysalis great for your early game in the jungle as it adds HP early and in exchange for 4 champion takedowns it removes the HP bonus and gives you extra AD.

The second one is more damage focused than the first:

PRECISION + DOMINATION +9% attack speed and +10 ability power or +6 attack damage, adaptive
Conqueror this is the other alternative in the Precision tree, this has more solo kill potential as it allows you to do some massive true damage after you stack it up, good for 1v1, basically for your own damage.
Triumph as I've said before, this is a great rune to enable us to get our items faster as it gives us 20g per takedown and restores 12% of missing HP.
Legend: Alacrity attack speed bonus which is well received by Irelia.
Coup de Grace damage boost on low hp targets to enable our resets, because why not?
Sudden Impact now this rune adds some sweet kill potential as it gives us magic penetration and lethality for a couple of seconds after we dash in with our Bladesurge.
Relentless Hunter this is great for roaming as it adds movement speed out of combat on takedowns, we can make use of this as a jungler as we're going to be roaming a lot.

Now the third build is all about burst damage and snowball potential:

DOMINATION + PRECISION +13 ability power or +8 attack damage, adaptive and +5.5% attack speed , we get more damage on our abilities and hit harder but less attack speed than the other builds.
Electrocute can easily be activated by 3 hits since Irelia's Bladesurge can easily be reset and you can hit multiple times with it. Adds massive burst damage early game. Now we have another alternative here which is Dark Harvest but this is for late game damage and I don't view Irelia as a strong late game burster, if you ask me you're better off sticking with early game snowball than late game damage as we're going to do the same thing with both runes anyway and early kill potential is better for your ganks.
Sudden Impact like I've said before, this rune adds more kill potential as it gives us magic penetration and lethality for a couple of seconds after we dash in with our Bladesurge.
Zombie Ward great rune for getting vision around the map as this will spawn another ward for a couple of seconds when your placed ward expires.
Relentless Hunter again, this is great for roaming as it adds out of combat movement speed on takedowns and since we're in the jungle we're going to roam most of the time.
Legend: Alacrity and Coup de Grace to finish off the build for more attack speed and damage.

Another choice can be Hail of Blades but that's for a more auto-attack focused build.

Irelia is primarily an Attack Damage bruiser/tank type of champion or fighter if you will ,with some burst potential if needed. She does in fact have some Ability Power scalings and can use AP items somewhat efficiently but I for one would advise against it as those builds are more for fun than for real ranked play.
Hunter's Machete and Refillable Potion are your starting items as these are the best for your early jungle low level clear. As your first item,directly after Hunter's Machete you want Tiamat as it's going to be the best for your jungle clear and auto-attack reset and we're going to be building towards Titanic Hydra anyway. After this you want Trinity Force, don't upgrade into Titanic Hydra before you buy Trinity Force because your damage and ganks will suffer a bit as a result.
For boots we have 2 choices Ninja Tabi and Mercury's Treads , we want Ninja Tabi if the enemy is more auto-attack focused than spell focused and in some cases we should consider Mercury's Treads, when the enemy has more ability power focused champions or crowd control abilities like a stun or a fear for example.
We have a lot of situational items for defensive purposes but our 2 main items out of the group are Dead Man's Plate and Maw of Malmortius, Dead Man's Plate being used when we need more defense is being needed and Maw of Malmortius is the item we want when the enemy has more magic damage and we want to kill the priority targets fast.
The best item out of the situational ones is Sterak's Gage as the AD and survivability it ads are massive.
Consider adding Adaptive Helm against champions like Kai'Sa, Teemo or Malzahar. Guardian Angel can also be a good choice which will save your life in some situations and it can be a game changer.
Consider adding Randuin's Omen when the enemy team has 2-3 critical items' users like Yasuo, Master Yi or Draven as it offers a nice critical damage reduction, armor, a good HP bonus and an attack speed reduction on enemies that hit you with basic attacks; it also has an active ability which slows around you in an area. Note: The attack speed reduction affects Yasuo's Steel Tempest as it reduces it's cooldown due to how the ability works with attack speed.
Consider adding Warmog's Armor when the enemy team has a lot of burst damage and lacks items like Liandry's Torment or Blade of the Ruined King, it is also a very good choice against Black Cleaver. Thornmail is extremely good against Blade of the Ruined King and auto-attack using champions like marksmen and even Yasuo.
Stormrazor is one of those situational items that you want only for the sheer amount of burst damage and kill potential it adds.

This champion is overall extremely good with little to no con's aside from the fact that it's a melee champion but the abilities it has fixes all our issues. I can't really say anything more here to be honest other than the fact that Irelia is one of the best champions out there right now.

- Guide finished and Updated for patch 8.13