I've changed my build since hitting level 30, and have found some of the flaws, even if they were minor. The way I have Irelia set up, you can steal lots of health early game, allowing you to get kills. I generally try to get kills at level 3, so I can dash, slow, and kill. Plus it's a great way to harass. People underestimate Irelia. Use that to your advantage.
Hope you like the build.

For runes, I went with Armor Pen, as they help a little early game in damage and a lot late game.
The other runes are used for survivalbility, for armor and magic resist.

I start off with a Vampiric scepter for early ganks, then building Trinity force. Trinity force is perfect for Irelia, since Phage gives you a slow, sheen gives you a 150% more damage when you dash to an enemy to hit, and zeal, allowing you to catch your enemies. Next, I rush Guardian's angel so I can be tanky to withstand my enemy by the time I'm level 11. After that, I build a bloodthirster for damage and life steal, get a Phantom dancer for attack speed and crit, then a infinity edge to top it off.