Tbh, Irelia is one of those champions that you can almost build anything on and be successful.
The general consensus is Trinity Force into straight tank items.
Personally, I like being well-rounded when it comes to Irelia, having a healthy mix of HP, Armor, and MR -- unless the opposing team is one-sided in physical and/or magic damage.
My build goes: Mercs > Triforce > Wits > Atma. The game usually ends around that point. I've found myself building more armor lately due to the abundance of Melee/AD, but it's hard to not have a decent chunk of MR, just because those specific items work so well with her.
- Mercs on top her passive make her very hard to CC/peel.
- Triforce is just good all around, she benefits from every aspect of it.
- Wits End for the additional tankyness/damage AND it helps you pump out more true-damage via attack speed.
- Atma's Impaler for additional tankyness/damge; plus it ties into a "piecemeal" crit build.
As for runes, they're obviously personal preference, but I'd suggest Attack Speed Reds/Quints, Armor Yellows, and MR Blues. Quints can be swapped for Movement Speed or whatever else your heart desires, but the more AS, the more damage you'll pump out with Hiten Style activated. Es op.
If you want to be more durable: Mercs > SV > Fratmas will definitely do the trick. Less damage and
burst, but not to the point where you're useless. You can stick on targets like nobodies business, all while being more durable. I'm too much of a sucker for Triforce on her, but as I've said there are many builds that can and will work.