This is a very risky build and should be used by people familiar with Irelia. I used it just to have fun and spice my game up and found it incredibly effective and enjoyable.

P.S. I'm not much of a guide writer but I'll do what I can.

Pros:

Amazing burst
lighting fast attack
ult that scales both AP and AD
amazing farming and pushing ability
1v1 / 1v2 pro

Cons:

squishy (you will be focused)
expensive
you have to play your early game right

Mark of Desolation for armor penetration. This helps throughout the game to help nuke those hard to nuke people such as Alistar or Malphite or other tanks.

Seal and Quintessence of Fortitude for the much needed early game health.

Alacrity for the attack speed which in turn will give you survivability (explained later)

Not gonna get into this too much. Building an offensive nuke instead of a tanky Irelia so build into the offense tree. I don't want to have to explain further.

Many people get stuff like exhaust and ghost. I don't care what you get as long as it works for you. Exhaust is great, but Irelia has either a stun or a slow with equilibrium strike and can easily chase with bladesurge. If you need that bit of extra distance, use flash and equilibrium followed by bladesurge and another equilibrium.

Flash is a must because if you charge in to nuke in a team fight you want to be able to flash out. You can grab some creeps quickly for some health and charge back in.

I choose teleport because I'm an American player who wants to be able to get back in lane fast. I back for my Sword of the Divine and I want back in the lane ASAP to pop a kill. If things go bad, you can tele to another tower and such.

I don't care if you use ghost or you use other spells. Grab ignite if you want. It doesn't matter as long as you get your items and your spells help you do so.

This build focuses on attack speed for the reason of pure survivability. With Hiten Style (also explained later) you gain health for every attack. This can be combined with life steal to give great survivability without using gold on armor or other things which take away from your damage output.

Start off with some damage and with Equilibrium Strike you should be good until you recall for either a Recurve Bow or Sword of the Divine. This is a great item because not only will it give you some of the faster attacks early game, you have the additional armor penetration you can activate and shred any early game items your enemy may have built up.

Next, grab a Phage to keep your damage up and give you a bit of health. This is sometimes where things can become difficult. By this time in the game, opponents will be stronger and you will have to mainly farm your butt off or get some great help with ganks or team fights. Here you will eventually be able to buy your boots and then buy a pickaxe as well for the Rageblade. Many people don't like the Rageblade, but for this build, it fits perfectly. Once you have it, your attack speed with be around 2 with 8 stacks and you'll be able to solo dragon if you want to because you gain health so fast with Hiten Style.

From the Rageblade you can build Sheen if you want, but I usually go for BF Sword as you need some damage behind your attack speed. The Cleaver gives you even more armor penetration to take down those tanky characters and gives you damage and more attack speed to boot. Finish up Trinity Force and an Infinity Edge and you should have around 2.4 attack speed with 8 stacks and do so much burst damage between your true damage, equilibrium strike, and ult to take down even a tanky champ like Alistar or Malphite in about 2 seconds (3 if they start running).

Another great thing about these items is that Irelia's ult now scales BOTH attack and magic damage. Items like the Rageblade give you both and are thus more effectively utilized.

For the skills:

You want to grab Equilibrium strike. It saves you when your running away (with the proper technique for a stun) and is one of your main damage outputs. Equilibrium strike dominates early game so grab it.

Next you want Hiten Style. This is where your entire survivability comes in. This build is focused on attack speed and every attack gives you a certain amount of health back. Mid to late game, if you loose a 1v1 or even a 1v2 fight with this build, your doing something wrong. The true damage also allows for burst nuking in a short period of time. You really want to max the amount of true damage you can do in the short time Hiten Style remains active.

Bladesurge. Its good for initiating and getting last hits (especially last hits), but I don't use it as a main source of damage.

Attacking looks something like this:

Bladesurge > Equilibrium Strike (assuming stun) > Hiten Style (true damage) > ult (if you don't have it don't worry) > another Equilibrium Strike > chase with Bladesurge and activate Hiten Style as it comes up.

Not sure how this plays out in ranked as I have yet to reach 30.....Let me know if anyone tries it, but I don't think it will work well because you will clearly be focused first and brought down.

Go nuke their squishiest character / their carry. Once their out you can redirect to anyone you want. You have a quick burst and if their tank is too far ahead, you can initiate for a quick second, stun, take 1/4 to 1/2 their health away before the fight even starts (your ult comes in handy). Try not to be the first one into a team fight, but Irelia can do an amazing initiation and burst. If you initiate first, you will probably die, so use all of your abilities and make sure you actually grab a kill as well.

It is important to grab everything you can. You'll be able to farm minions effectively using bladesurge. Get into the enemy jungle and steal some stuff. Grab dragon. Your build is expensive, but it's worth it.

Try it out a couple of times before rating. If you don't get the build down the first time, try it again in a different game. Once you get the hang of what you can do, the build becomes way more efficient and you can burst more effectively.