So,Irelia isn't one of the more popular champions. BUT nonetheless she is a great hero and one of the best ad carries in the game, much like Yi.
Anyway, lets get started. Irelia played wrong at the start is usually going to look very very under powered. So let's make sure you're starting out right.
Strangly enough Hiten style may not look so great since it IS white damage but regardless its a huge amount of burst with EQ Strike. You'll really notice how much damage the Hiten style active is once you start using it =P.
Keep in mind This is a DPS build.. So its more glass cannon then it should be..

Maybe add a few survive items in there in place of some of the dps but This is pure dps.

Items!!! I go for Sheen first for the extra burst damage early on with EQ strike, the mana and hiten style, followed by Berserker's Greaves for the attack speed and Wit's End also for the attack speed. Since your early game damage is going to come from hiten style this combination works very well.
After this go for a Youmuu's Ghostblade the armor pent and cool down reduction is very nice also the crit and damage. Ghost blade is going to really burst you up there with damage after sword of the divine.
After that go ahead and finish off Trinity Force for the slow on phage and the extra crit / damage, and extra damage from sheen proc. trinity force will probably be mid game maybe late game depending on how the games going. Regardless Infinity Edge will be the next best item increased crit damage and crit and lots of damage.
After all of that get Madred's Bloodrazor for extra tank damage a bit more burst of speed. If you actually manage to complete the build its probably a rough game. Either way this is the combination i have found to be the most damage.

Runes and masterys are pretty standard for an AD. When i play i find runes and masteries more of a preference then a MUST or something like that.
Also Summoner spells again more preference. I choose flash for getting in range of charge or EQ strike. because she already has the life steal and a slow (possibly a stun) a charge and large damage dealing potential

Ghost as well for chasing or running just whatever.

Irelia is NOT a tank in ANY way. A lot argue that she needs survivable because shes squishy. but i mean seriously? she has the MOST life steal in the game next to WW. She is a assassin / carry late game whatever. Either way =P

Edit: fixed a misplace sentence.

Please try this build and enjoy it. This is such a great build. I have played irelia for quite a while and haven't found many if not any... flaws with my build pretty much just staying alive is the only problem and its really not that bad if you play quickly and get in and out when needed etc.. Please vote and leave responses below