Hello there, I just want you to give u a few hints how u can(!) play Irelia.
At first I wanna tell you, that Irelia isn't a Champion who kills their enemies extremely fast (like the Ninja-Assassine Akali) but she kills her enemies.
She is really usefull in teamfights and she can farm like hell!

+ awesome farmer with her Q
+ constant DPS
+ isn't that squichy
+ if u know how to play her, u can dominate the game (especially if u have a mate on your side)
+ can restore health while running away from enemies with her ultimate (awesome if you have Bloodthirster already)
+(Jungle) She can really good gank!

- needs to farm to 'join' the game => if the enemies won't let you farm it's almost a **** game for you :)
- mana problems at the beginning
- hasn't got an awesome escape mechanism like Corki (for example)

Recommended Runes

Armor Pen is good in early (and probably late game if they aren't stacking a lot of Armor) for
- Farming
- Going for firstblood

Flat Armor for more surviveability during the early/mid game

Magic Resistance/lvl for more surviveability during mid/end game

Flat ArmorPen(Quintessence) is combined with the masteries + Marks really awesome (31 ArmorPen at the beginning) your enemies will almost have around <5 Armor at the beginning

Optional Runes

Instead of taking Armor for seals, u can go for Dodge
And instead of taking MRes/lvl glyphs, u can go for CDR/lvl to 'spam' your abilities during mid/end game, it's just awesome if u can use your Q really often in a teamfight (especially with the bloodthirster in your bag)

Jungle Runes

I prefer takeing the 'normal' jungle runes.
ArmorPen Marks, Armor Seals, Mres/Lvl Glyphs, ArmorPen Quintessences (u can also go for Flat HP)

Recommended Summoner Spells

Flash is (in my opinion) the best summoner spell in this game, it can be used for chasing or escaping (especially over walls) or just to show the enemies, that u are the boss :P

The second summoner spell (Ghost) is also super, because u can get out of ganks, u can faster join the teamfights and u can chase the enemies down (if u need to)

Optional

Teleport is a really good choice if you think you don't need Ghost. You can easily recall and buy your first items without losing XP/Gold but attention clean your lane from enemies minions, so that your turret doesn't get any hits and that you don't lose XP/Gold.

Optional Summoner Spells

U can switch out Ghost for some other Summoner Spells:

Ignite: Just if your team won't take Ignite and u want to have it (e.g Jannas Ulti) or if u want to see ur enemy dead

Exhaust: Trinity Force has got a slow, this is just usefull before getting Trinity or for escaping, but u can just slow one enemy down (so for escaping Ghost > Exhaust)

Teleport: Awesome if u want to dominate your lane or to defend your turrets. U can easily port back , shop your items and go back. Really usefull. I Prefer Ghost, but it's up to u.

Not recommended / useless Summoner Spells

Clairvoyance: Let the supporter take this summoner spell!
Clarity: Probably good in early game,useless(!) in late game.
Cleanse: No need with ur passive + mercury's treads.
Fortify: Uhm no, let the tank / supporter take this :D
Heal: Never take this!
Rally: What's this?! Don't think about it!
Revive: No no and again no!
Smite: Just if you go jungling

At first u max up your Q, it's your main damage source during the whole game. U can switch between W and E at lvl 2, depends on
- how aggressive your lane is
- which enemies you face
Later after maxing your Q, I've decided to max both abilities at the same time.
1. The passive from W combined with your ulti can heal you a lot of times for 50% (shooting it through the next minion wave or through all the enemies)
2. U need to reduce the CD on ur E. Imagine, u want to escape (u have less life than your enemy) and your stun isn't ready or (if he has more life) u can't slow him. It safed my *** so many times!

Route

1) Start at blue. Using your W during the whole fight, smite at the end (don't forget your Heal Potion(s)
2) Go on with wolves, using your W everytime it's ready and your E after an autoattack (it won't reset your auto attack CD)(+heal pot)
3) Move to wraiths, waiting a few secs till Smite is ready again. Smite the big blue one and go on killing the others with W and E (same 'tactic')(+heal pot)
4) Go to the double Golems and kill them with W and E(+heal pot)
5) Finish your first route with the red buff. If you have too long CD on smite, wait a few secs (till ~5-10secs rested). Go on killing it with W and E and smite at the end.(+heal pot)
Now it's up to u. If u think your lanes need help => gank, if not go to the wolves and go on jungling.

Items

Start with the 'normal' jungler Items.
If u aren't needed for ganks rush Wriggles before Boots, if u are needed => boots.
Later on, trade the wriggles with a bloodthirster (not so necessary at the beginning, 95% of all games u don't get the chance to do this, but just for the case)

Now to one of the most important parts, the item choice.
I've seen many Irelia Builds, from building up as pure DPS (Infinity, Phantom, Black Cleaver) up to Pure Tank (just with Trinity Force). This guide is an offtank guide. You get AD(scales awesome with your Q), lifesteal (awesome for Q and Ulti => restoring health), Health (Trinity Force, Banshees, Randuins) and good resistances (armor / mres).

Core Items

Damage Items

You should try to get these 2 Items(Trinity + Bloodthirster) ASAP(as soon as possible).Wit's end should bought as the 5th / 6th items, it's up to you! If you think ur 'too squichy' then skip the second Bloodthirster and get another tanky Item.

Tank Items

In my opinion,GA is a must have! The Resurrect is like a 'OMFG she resurrects, RUN!'. With this Buff up, u won't normally get focussed if yes, u restore health like hell (Lifesteal and W passive) and even if u've died, u just resurrect and kill them all. Normally they don't have any CC anymore ready, so it's easy for you to restore health.

Examples for Itembuilds

Against a lot of squichies

Against Offtanky / Tanky

Balanced Team

High DPS Team

Early Game

U have to take a solo lane! Last hit nearly every minion, if needed with ur Q!But attention, don't waste all of ur mana, u probably need it for getting a kill or for stunning + running (nice rhyme huh? :D)
Around lvl 6-8 u should have finished your Philo Stones, Mercury Treads and if u are good in last hitting your heart of gold

Mid Game

Now you have a better mana regen with your philo stone, u are fast enough to escape ganks / join ganks. But while entering mid game, the game gets more difficult. U should go on with farming but u shouldn't let your mates die because u aren't there in the Teamfight, the right balance combined with map awareness is the key!
If u see everybody is missing, inform your mates to regroup somewhere or to go back to their turrets (to negate possible ganks).
U really have to get last hits like hell, the reason is, u need the gold advantage! If you have boots + Trinity Force + Bloodthirster ready (and ur mates + enemies just have finished boots + their first Item) then u've done well your laneing phase!
Now it's time to finish ur lane or to gank, choose enemies like Ashe to gank (she's easy to get) and inform your mate that u are comeing. Don't worry if u 'just' get an assist, u are playing in a team, so the kill for the team counts, not ur own score for maximizing ur ego!
Don't forget, if u can't get any kills, GO ON FARMING!

End Game

If the teamstats are balanced (for example 30 for ur team and 28 for the others) then it's time to turn the game. U should have farmed like hell and u should have get a few kills / assists. This gives u the best advantage u could get in this game.
U should have Boots, Trinity, Bloodthirster,Guardian Angel and probably your FoN. If u haven't 1 or 2 items more than ur enemies, than u or ure team have done something wrong!
If u have more items, change the game! You can initiate the teamfights (if you are the tank or even not, it doesn't care, u should be tanky enough)! Go out for their Carries => stun / slow him/her. So your carries will have these few seconds to deal a huge amount of damage, their carries MUST drop, but this shouldn't be a problem (just if you don't have totally braindead people)
If u think it sounds easy - yeah it is. U have a lot of damage (every ~4 secs ur Q and E followed with W (Autoattacks) + Ulti = great damage!) combined with ur surviveability = awesome!
If u are solo queueing, then it's probably not so easy to ace them. Care about your supporter / carry. Stun / slow the attacker if it's necessary, get on the enemies nerves. U can play a bit like a ninja assassin :P

If u read this all, thank you for taking the time to read my guide. I read several guides for Irelia and I was always thinking that they aren't showing me the playstyle I'd like to have on Irelia. Pure DPS guides are good for insane damage on your Q. I've tried them out several times and it wasn't the playstyle I prefer. Also the pure tank guides are **** (in my opinion). At least I've found a way to play Irelia which is the playstyle I like. She can tank a lot of damage and if it's needed (ok, of course it's needed sometimes :P) u can restore easily your health with your Ultimate. She doesn't feel like a ninja assassin, but also not like a boring tank, it's just Irelia!

I hope u enjoyed my guide and item build.

Don't forget to vote and leave a comment!
PS: I don't know how to put the Ability Icons / Summoner Spells Icons into the text, if i find it out, I'll edit in (it will be easier to read!)