I know what you are saying, AP IRELIA THATS DUMB!
-She has more abilities that use ability power to deal MAGIC damage, so with proper penetration, these spells do a heck of a lot of dmg for your purposes.
-Not only that, but magic penetration will help her Q because it applies ON HIT effects, like:
-->Wit's end
-->Madreds Bloodrazor
Therefore Her Q benefits more from magic penetration than armor penetration, as:
Madreds Hitting a very low health target champion of2000 Health does 80 damage a hit, what i would consider the minimum from MBR's damage added to one Q.
That is equivalent to a Halfway charged Bloodthirster or An infinity edge both of which you would never invest all the money to buy with 0 survivability to them.
-->Not even the point because MBR is giving you Attack speed for more healing and W spam.
Therefore, with those two magic dealing-tanky on hit items, I give you:
IRELIA THE PENETRATOR

---

I dont think there is any alternative to the MPEN because it is what this build is all about, with your damage coming from 2 Magic damage dealing on hit items, you need this unless you want to build haunting guise or get sorc boots.
---I have some Flat AS and mana regen for early game laning, so you get as many W harass hits as possible.
Mix this with regen/level, you dont need a major mana regen item to spam your whole rotation on someone, which is guarunteed kill.
---CDR/level so you have farmed to 18 and are ready to just dominate, brings ur W up a lot faster, more chasing too, more stuns, more ults.
----Move speed quints so you can ignore boots and be faster than people, buying your 2 on-hitters and being able to stay on people to W them to their grizly fates.

This setup just seems to work. Remember you are not trying to crit or pierce armor, but pierce magic resist, or support your W in any way possible. Sadly some of these must be spent so you can complete this mastery tree for healps of damage.
If your feeling skittish maybe try a Defensive buid, with added AS and AP!!!.

You are going straight for amazing damage.
Q applies all on-hits//gets you where you want to be at level 1, that is all you need.
!!!!!!!!!!!!!!!!!!!!!!
W-THIS IS THE BOMB FOR ON HIT BUILDS
--kayle/kogmaw (2 great on-hitters) have abilities that shred some magic resist and armor.
>They lack the amazing amount of damage this W adds.
--This ability adds 75 UNNEGATABLE damage to each of Irelia's attacks. Amazing at the 1.5 AS you will achieve here. That is 112 extra damage a second, no stopping it.
--It also heals you, giving this build a lifesteal feeling, healing you for ~45 health/second when you are in the fray, this health made valuable by the tankyness of this build.
!!!!!!!!!!!!!!!!!!!!!!!
R- This baby will hit HARD with your masteries and runes (And end game penetration), Therefore healing you in large amounts

E- Stun people whenever you are lower than them. Also made into a quasi-nuke by all the magic pen you have, but more importantly a slow/stun so you can destroy someones health bar or get a carry focused by your team.

Q applies all on-hits//gets you where you want to be at level 1, that is all you need.

The bread and butter of this build are rushing your wits end and madreds through FARMING or just getting kills.
THEN:

You want some boots. I just highly recommend Merc treads because they are so broken on Irelia with her passive. Hard to pass up walking out of a galio ult half way through. Or negating 3-4 stuns with Passive+mercs/QSS/Banshees. This build does so much damage, it would be dumb not to get it all out there, so reducing CC is the way to go. No other boots really do it for her, except:
Tabis---Recommended only if you have nimbleness, and defensive masteries,and the enemy team has a ton of autoattack champs (NOT AD MAGES LIKE RENEKTON) can save your life from a third silver bolt by vayne.

Afterwards, You should meditate to find out what tanky items you need to help you put out the 2 on-hit-item-and-W combo onto whoever you want, therefore killing them.
Basically, you can see other on-hitters machinegun some pretty good numbers of damage. But it is just nothing like Irelia's, where besides wits end, everything number appearing is either 75 EVERY TIME or some number in triple didgits. The damage this build outputs is just unmatched by anything I have ever seen, from buying:
-Wits end
-MBR(Madreds BloodRazor)
-Abyssal Scepter
and after that its basically building whatever you can imagine a very fed irelia could want, in the form of two items. I suggest:
-Defensive items. Irelia is a big name in tanky DPS, for many reasons (a majority of them being her dealing true damage with each hit from her W :P). So please, take advantage of an excellent oppurtunity of the 2 best defensive items that will just own their team.
-->Frozen heart: Best aura for this build since you are trying to out-DPS some of the biggeest DPSers in this game. Especially when lots of AS is being purchased by enemies. Will help you stun them more and catch up to pesky runners, as well as snipin em with your ult.
-->Banshees Viel: When you face lots of annoying CCs from afar, if you find you cant penetrate their ranks.
-->QSS-2nd to banshees (though should be purchased for the same circumstances), due to lack of health or mana, but amazing active if you really just get one too many 50% decreased duration crowd controls and need to rip through them like a baws.
-->Spirit Visage:Great for tanking towers for ridiculously long with your insane AS and W heals. Also great (to get your vamp on) versus lifesteal champs like akali or jax that usually do a lot of magic damage as well.
-->Randuins Omen: For very fast, agile opponents or ones that stick together in closely knit groups of squishies. Consider Frozen Heart first for this build b/c CDR is what she needs to keep W going and slow/stun in a constant stream from equilibrium strikes.
-->Force of nature:If you want to go superman and be so fast it almost looks fake, get this. Consider Banshees veil first because you will have plenty of MR, u dont need this huge chunk of it. (Which has nothing else to offer.)
-->Sunfire Cape:Great with the magic pen, Has armor and health for going against AD bruisers who are just barely out Tanky-DPSing you.
-->Atmas: Still legit if you wanted a warmogs or something as the other open slot. (or, perhaps, ROA? O.o)
-Auras to help your team: Starks, Frozen heart, Aegis of the legion...etc....
-If you are going to build these as damage slots, do something that will benefit your heal from W and the true damage. THAT IS MANDATORY IF YOU ARE NOT BUILDING DEFENSIVE FOR THESE ITEMS. Something that adds at least 20% attackspeed.
Go by this rule and if you dont need the tankyness, people will literally die in 2-3 seconds of you being on them.

Examples of useful attack speed items:
Cloak and dagger
Starks Fervor
The Black Cleaver
Malady
Phantom dancer (instead of boots if you want to--super effective with the movespeed quints)
Not so bad but not out of this world items:
Guinsoos Rageblade- Dont know why you would unless you really enjoy your ult over defensive items or wicked DPS.
youmoos-Too much of a midgame item, and you already dominate mid game extremely hard. Active would be nice, but armor pen is not for this build. CDR can be found elsewhere.
Nashors-CDR good, manaper5 good if you want 'extended engagements'. I prefer to just kill people in a matter of seconds then get on with my day, instead of skirmishing, which this item would be for.
SOTD-Armor pen not useful, but good magic damage, only not too viable because no one really has insane dodge anymore. But if you see just one too many dodge procs, maybe it is time to make your W slice through this tanky SOB with the tabis.
Triforce- Not bad because this build keeps you near your base attack damage, but the recent sheen nerf just makes this item suck for everyone :*(.

If you get a third attack speed item, my recommendation is this--WALK UP TO A TOWER AND BEGIN WHACHING. WHIChEVER SINGLE TOWER YOU WANT. YOU WILL GET IT DOWN TO ABOUT 40% HEALTH, 1v1, from full (especially with an ally) <3 Irelia's W

If your enemies have hella magic resist(2-3+ items), consider a Void Staff. With this item, Your ult, E, and on-hit items will remind you of how hard you were hitting when you first got your wits end.

Pros:
Outdamaging everyone. Only way to say it. There is no defense against this build.
Tanky, Very versatile.
Fun because almost no teams will leave you baffled as to what to build to annihilate them.
-find out what is killing you, build an item to counter it.
Surprising. People just dont know what to think about this build, no one has ever seen its entirity before.
Ultimate fighter-Jax-par turnarounds, 2v1s, WTF I LIVED WITH 3 HEALTHs, early solo barons and multikills.
Turret dive fiend- I am diving nasus's all the time with their ult up because i can usually secure a kill. Can tank towers for soooooooo long, as long as you keep hittin' stuff.
Unbelievable-Sometimes tanks literally melt faster than squishies from this build.

Cons-
--Must be in combat, hitting a lot. On-hit items aren't RIDICULUSLY tanky. They will keep you alive very well, but you can get yourself in a really bad position sometimes trying to carry your team.
--Must be quite farmed and/or fed to be truly dominant. Requires a lot of farm, though on-hit items ARE cheaper than most. Lets face it, without your MBR and wits end, your an AP Irelia :(

->this probably means, if you cant just find a way to AFK farm/ get some legit last hits, you will be picked on and targeted, and weak except for your normal just being Irelia. (OR get kills, which you should be able to do with a normal Irelia before trying to learn how to use this build.) You are not amazing quality DPS because of your runes, even though masteries do compensate a good amount.
->THEREFOUR patience must sometimes be a virtue to make your tower do the work for a kill. :(
--Static (in the very early game)-Hard because you have 2 starting items you really HAVE to build, which could result in some problems if income is low or you are far behind and need to substitute in an item.

You really dont need your team, as you can pretty much take anyone 1v1.
-->so why not treat it like a toy army and outfit it as best as possible, with item auras and buffs or debuffs for the enemy. If you do this then you should be able to jump where the &%$^ you want when you want to. So you can commence to chainsawing.
Hacking people up in LoL is besk done 1 by one, so use your flash bladesurge to get to someone.
If you continually try this, and catch people off their guard they will become scared of you.
Fear is the best possible thing you can have and it works the same way as fear in lol.
MAKE SURE you are the lion in the grass always stalking the pray that will scatter as if they were hit by a level 5 terrify.
---If your any good at fishing or somesuch, you want to not be seen until you want to be seen,
-and when you jump out, make them scatter, find the one that went a wrong way.
--Just walk in there and make sure to stun someone who is just about to do a lot of damage or is doing a lot of damage.
--done properly, you will have jumped straight onto someone, attacking with your W and heaing, but losing health so you can stun someone. An Irelia with full health is USELESS.