I typically go Boots/Chain and Cloth -> Phage -> Merc -> Warden's -> Triforce -> Wit's End if MR needed, Kitae's if HP stacking, Black Cleaver if none of the above -> Randuin's -> FH -> Finish with WE if not used before or counter any last whisper/void staff on the map with a perspective mitigation stat.
Play is usually more important for me than items. Hiten Style hurts, but against targets with a lot of HP it's going to take too long to chunk them. Go right for the squishy targets like an assassin (which is really how your kit is setup) and make sure to prioritize smart engagements. Always have an escape plan if your HP is falling faster than their's is, wait for W/E to go back up, and reengage. Playing Irelia is basically a battle of attrition for engagements since so much of her damage is DPS based. You need the mitigation to survive that initial volley of burst, at which point you can E and quickly start to wear down the enemy target. Her kit is incredibly adaptable to the situation; if you're getting fed faster, rush right into more damage. If your team needs a brick wall in the front, stick to TF-Wit's and build up tankiness. You still have high kit damage even as a tank and lasting longer in a fight just means more hiten style strikes.