This is my guide to playing a solid steady dps, and off-tank Irelia. Irelia was the veryy first champion i had ever wanted to play while browsing the LoL database waiting for it to download. Though her recent nerf has drastically affected her play style, she can still be used effectively to deny the enemy team. She is a snowballer, such as vayne if she does well in the begining she does well endgame.
Bear with me as i introduce you to Irelia, The anti-carry.

I choose to go with armor pen quints, and marks, these will help your damage early game agianst those long ranged laners who are just oh so squishy ;). Seals i chose magic resist per level, to scale endgame with some of the casters and add that bit of survivability. For glyphs they are CDR per level, her bladesurge can be taken down to a 4.9 seconds with masteries/runes, a.k.a uber farm.
As for masteries the extra armor will help to block the other teams armor pen runes, and adding damage to your early game as well.

I personally always start off with a scepter of lifesteal. What more could an Irelia possibly want than to be able to farm as long as she pleases. Get this and your laning phase just turned into a laning episode. If you are 5v5 farm until you have enough to get your sword and cloth armor for madreds razer to be turned into a wiggles, this item offers everything irelia needs, armor, lifesteal and attack damage, and will greatly help you in the lane and out.
Next you deffinetily will want to begin your Triforce, the main item in most irelia builds and for good reason. Everything on it scales with her and her abilites. I personally start with saphire and make sheen then move to zeal then phage, as that boost my lifesteal and damage more effectively. So now that you got a bit of damage, and some armor whats next you ask? Make yourself solid, and slap a chainmail vest on this girl and very few ad can take you down early game with wiggles and your passive/Tri.
Follow up to a wardens, then randuins and consider yourself a tank already, at this point you should be shutting down your lane, or retreating to safer ground to farm more (ill explain later on her laning phase).
On your way back to base pick up a pair of boots, and i have no preference but i tend to do with zerk greaves, or if i need that bit of extra MR i get treads, or possibly ninja tabi all boots on all champions are situational and personal preference IMO.
Now we need to start doing damage, this may be late game or near ending but grab a brutalizer and farm that thing into a ghostblade asap, and if your doing that well slap on an inf edge and consider your sheen activation some of the biggest melee burst youve seen.
Other items besides ghostblade is possible to grab a phantom OR a black cleaver or blood thirster. If you want to deny there tanks take the BC, if you want to live longer with some more burst take the BT, and if you want to eat buildings take the phantom. Ghostblade is iffy for me as it probides everything she needs, but the damage can sometimes seem meager, so some armor penetration damage math will be needed for that.

Early game- You will want to pick up a laning partner that is ranged or at the least can harass well while you farm minons. After grabbing your scepter head to the lane and sit passive until the minions come to you and last hit repeteadly with your abilites. I start out with e most the time simply because i prefer to be cautious of gganks and the slow/stun helps to initiate/run away/save teamates life. grab e,then w, then q, and max w then e. Why equilibrium strike? simply because your aiming for squishes hopefully who have low MR and armor stacks level wise and raises most MR does not so its better for early game fighting.
Your W hiten style truly makes you a laner that you cant ignore. I personally after wiggles have sat in the lane with a leblanc and cait harrassing me repeteadly and i did not move because i healed far too much to be bother by 6- level skills at the time xD.
Keep last hitting with your Q on minions. Harder than it sounds, when the minon is at just below 1/3 hp you should then Q to that minion E the next one and Q agian. Mana efficient? No. Reason being-Irelia doesnt suffer from mana deprivation. (side note on ult) now that her ult is no longer a "wtfijustfarmedallmyhpbackfromthatminonwavenomnom" it is now more damage less healing, but more damage=more healing. Dont be afraid to use it ever, use it to harrass champions behind there turret just face them and put your mouse on them xD.

Mid-game - At this point most of this game is at team fights/ganks. Your job? Scare turret opressing *******s from the opposing team with your ult and clear minon waves with it as they hit the tower q/e/w autohit and back the fudge up. Team fights i aim for anyone,except the tank, but mostly aim for there carry then the squishes in the back of the line, they go down hard and you eat any AD burst like candy.

End-game - you can easily take turrets. by yourself. with two or three people attacking you. Your W/R when using it at a turret activates Triforce= three shot a turret. Lead your team intitiate and use your ult to defend any turrets or buildings from harm.

Pros- Easily farms.
huge carry/dps tank endgame
Jungling is very possible
not easily cc'ed
Great burst damage

Cons- Bad early game/low damage
If not fed, not good/item dependent

You may be thinking, "uhmmm, you use an ability and you hit harder after". Yes, and no. Triforce and sheen have a 2 second cooldown between uses of each. YOu will want to time these perfectly. The reason Irelia does so well with this item is the ridiculous amount of R spam she can do, within ten seconds she can do 500% more damage from sheen/tri than normal. Add in QWE and thats 300% more. DOnt just go at an enemy Q/E/W unless go for a killing blow, save E for when you can stun or need to finish them off. Space your R aparts to activate sheen, unless in a team fight with massive burst. Learn to do this correctly and all of a sudden your damage got better.

Why AD and not AP? Simply put, Recent nerfs have made AP Irelia simply unviable, and lackluster in her steady dps. Its minor burst, and a laughable amount of damage from her ult.
Why more AD than AS? "But her W!!! It will keep her alive"
Well W is more used to restore your hp and add that sheen activation and bit of true damage for tanks. But really with a triforce in hand and no AS with just an inf edge she can burst so quickly and hard that it would seem that you have a lot attack speed anyways. It helps yes, dependant on how you decided to follow this build. (ghostblade,phantom,BC,BT, etc.)

I tend to get E first for the slow/stun, W second and then Q third. Maxing out W then E, and ult whenever neccesary. Q's nerf now scales badly in the begining and mid game, but abuse Q endgame it is massive damage.

Optional really. I prefer ghost and teleport to port to a turret with huge wave of minons, and farm farm farm, not to mention teleport+ult= save a turret just in time. Flash, exhaust, and cleanse < are most deffinetly other viable summoner spells.

All in all, Irelia can be terrible, or can be amazing it all depends on her farming, early game laning phase. In the correct hands she is devastating despite the recent nerfs to her </3.