Not Updated For Current Season
So Irelia is broken right now.
She has some odd mechanics, for instance her Q when used behind a wall does the damage but doesn't teleport her. This could be working as intended but it feels odd.
The first build is a laning hybrid build that works as well as any build I've seen for her so far. It is however entirely dependent on your ability to last hit with you q. Spam it, but only spam it if you last hit with it on creeps. If you suck at that then you will be getting all your items late game instead of mid.
So an explanation of the second build. It is attack speed and crit based. Buy the items in the order I specify and remember that you should start as a jungler, and then when you are out of the jungle (level 4-5) go into pure gank mode. If you go out at level 4 make sure you have your stun/slow instead of the extra point into your Q. Personally I prefer to gank at 5.
Bear in mind as well that the secondary build is expensive which is why you are buying scattered bits and pieces of different items as you go along. But when its all said and done you will be a walking *** kicker.
Added a crit/attack speed build. It feels like it works well but its very expensive. So if you are sucking in a game don't go that route.
**NEW CHANGES 0.2**
So a friend of mine suggested hybrid build. I called him an idiot and he played a game with it. It worked. I tried it. I loved it. I'm not giving him any credit for clearly making the build, but ****, its good. Try it out. It's a laning build No smite, just exhaust and ghost.