I'm not a famous player, nor do I play at a high elo. I just made this build guide for anyone who wants to try Irelia out. I probably won't be updating this guide so at the time this was made it was 1/30/13.

I recommend Ghost/Ignite.
Ignite- Either ignite/exhaust are the best offensive summoners in the game, there's no problem with exhaust I just prefer the true damage over time from Ignite to help me kill my opponent. Works extremely well against Vlad/Mundo top.
Ghost- I like ghost cause it helps me escape and chase. With all the tenacity you get from Merc treads/zephyr, masteries and her great passive, you're unstoppable from sticking on the enemy carries. There's no problem running flash, Irelia has no built in escape so its very useful for escaping ganks.

Short item explanation (I'll only include items I feel Irelia will use)
Mobility
Ninja Tabi - Grants Armor and reduces 10% damage from basic attacks. Works great against enemy AD carry.
Mercury Treads - Grants MR and Tenacity Works great against AP carries/cc.
Offensive
Zephyr - Grants AD, ASPD, MS, and Tenacity, works well against CC.
Trinity force - Tons of damage
Iceborn Gauntlet - Sheen proc with AOE Slow, grants mana/armor/cdr.
Wit's End - Added magic damage on hit, grants MR/ASPD.
Blade of the Ruined King - Added phys damage on hit based on 4% of enemy's current HP, heals you for half that amount. Grants AD/Lifesteal, Active-Slows and steals movement speed+HP from the enemy you cast it on.
Defensive
Guardian Angel - Grants you an extra life with 30%HP/MP - 5 Minute cd.
Frozen Heart - Grants armor/mana/cdr. Passive slows enemies' attack speed around you by 20%.
Randiun's Omen - Grants armor/HP. Passive slows 20%attackspeed/10%movement speed of enemies that hit you.
Warmog's Armor - Grants HP. Passively regenerates 1.5% of your max HP per 5 seconds.

The main reason I made this guide was to share with everyone my item builds.

First back- Doran's blades help a lot, consider buying them if you can't buy any really big items. Remember boots were nerfed so they aren't exactly necessary anymore. If you can just gear up for your phage. Grab 1-2 wards and some potions.
Second back- Finish up phage, grab boots work towards wit's end/stinger based on enemy team, if you need sustain grab Vampiric scepter. ATTACK SPEED IS VITAL FOR TRADING! Hiten Style will hit like a truck. Grab a couple wards, even if you plan on roaming you still need to know when your lane opponent comes to roam as well.
By this time you should have Phage+Wits/Stinger, don't finish up your Zephyr, start towards BotRK.

My core build would be
Warmogs, Blade of the Ruined, Triforce, and Guardian's Angel.

Ruined King scales well into the late game and since lately everyone has been building Warmog's it fits perfectly. Warmog's just gives you that huge HP pool you need to be a tank. Triforce's passives are amazing. GA work's well in MOST cases but not every case sadly. I still like it in the core though just because it gives both armor/MR and the passive with warmog's gives you a guarenteed 1k hp revival.

Against Double AP -
Mercury treads and Wit's End with the core build. (No need to remove anything from the core build, everything works nicely.)

Against Heavy AD -
This is where the build may vary. Even the core build can change up here because we hate AD's that much ^^. Tabis + Zephyr, giving you the cc reduction and attackspeed+MS to stick on them. Frozen Heart replacing GA, you need the FH passive more than the GA in this scenario. Omen's MAY replace GA or Warmog's now that it gives HP. Omens+FH makes you a nightmare for ad carries and ad casters will have a hard time taking you down. Omen's passive+active help you and your team while Warmog's helps mainly yourself. But by helping yourself you kind of help your team in a sense where the longer you're able to be in a fight the more help you'll be.

I'm just dropping this build off for anyone looking to find a short explanation of what's good for Irelia, I didn't include many items such as Ghostblade, Black cleaver, Bloodthirster because I prefer not to get those item's on Irelia.

Leave comments, helpful tips for other viewers (maybe a tip not to follow anything I said in this BUILD GUIDE haha.. :c)