Im pretty new with irelia so will not pretend im so good with her, but Is�´ve had some success with this build anyways.The big question mark for me was which skill to level up first because all of them get quite much stronger then the level before. Blade surge get really low CD on max lvl, and Hiten style makes u really relly high damage fast, as well as getting staying power.
The last, Transcendent Blades, it�´s Slow/stun gets longer.
I noticed that is depends lots on your playstyle but after playing a while, as hiten style makes u quite strong, it benefits well from attack speed, and u dont have so much of that early on. And as Blade surge can refresh, I had success with Transcendent Blades. The real let down with this skill is that is scales with AP. But I recommend u try out which skill to max out first as they are all god skills to first max. Now I have just spread it out.

BUILD 1
This is the only build I have tried and honestly i just got some inspiration from several different builds on the site. I prefer Armour pen Marks. yes, she have True damage skill, and lots of her abilities are magic damage, but i want her autoattack to hurt alot even though that.
Boots. One champ that doesn't is in need of Merc Threads as much.
If u skill upp Hiten Style first,Bersker greaves.
If not, Sorcerer's is a great option.
Then Sheen and Hexdrinker first, as it makes her very viable ganker early on. And people do underestimates the Hexdrinker + her passive combo (and the rest of the skills after ofc). Just like that, irelia is in your face, with 200 hp left and when u thought u were gonna kill her in your stun/runeprison/slow
or whatever. Hexdrinker is at its stongest early on.
Trinity Force is given on her, because u want to jump their face and stomp it quick.
Every single aspect on Trinity Force synergize very well with Irelia.
Infinity Edge ,makes u stomp even harder,as you want to stomp away their face and get the hell out of there.
Zeke's Harbinger ,It synergizes very well vith her. Nothing more to say. As Trinity Force it is really good on her. and gets you a bit more staying power in the teamfight that should be going on by know.
But as the last item, it is a more difficult decision. I like Bloodthirster,
but some people prefer Banshee's Veil, or Guardian Angel. I like the extra damage, and just hopes that the lifesteal is enough to keep me alive.

The great thing i found about this build, is that it is not overly expensive early on. U get quite strong quick, with Sheen and Hexdrinker, and if u are quick to use that gap when u are a bit stronger and gank and kill some, u can get ahead and get Trinity Force and Infinity Edge and make it hard for them to catch up.