Through out my year or so of playing League of Legends I have never liked a champion more than Irelia. The versatility of this champion is amazing. Playing her right can really pay off and I hope to help you do just that with this guide.

Armor pen marks are the best in my opinion since most of Irelia's damage is physical and both Bladesurge and Transcendent Blades deal physical damage as well.
Armor seals and Magic Resist glyphs are the best early game to give you that tankyness before you start building your items.
The move speed quintessences are just to be able to harass in early game lane.

I open with a Regrowth pendant and health potion. If you are new to Irelia and have trouble last hitting without using abilities you can switch the health potion for a mana potion. These items help me stay in lane and farm while being able to take some of the damage from enemy champs and minions without too much of a sweat.
The Philosopher's stone is mostly for the gold per 5 and health per 5, but the mana regen makes it so you won't have to buy mana potions or "B" to replenesh your mana.
The heart of gold serves pretty much the same purpose (gold per 5 and health. not regen.) as the Philosopher's stone except you will be keeping it to make a Randuin's Omen later.
I know it seems redundant to get mercury treads while Irelia's passive does pretty much the same thing, but in teamfights and chasing if the enemy hits you with any cc they have just wasted their cooldown.
The Trinity force serves many purposes. It gives you a little of everything: mana, health, ability power, attack damage, attack speed and critical chance. In addition to the stats it gives you passive abilities including increased movement speed, chance to slow your target on hit and increased auto attack damage after using an ability(this passive works amazing with hitten style).
The Force of Nature not only gives you an incredible amount of health regeneration, but it gives movement speed and magic resist to help with tanking and chasing.
Randuin's Omen is a great item in many respects, it gives health, health regen, armor and cooldown reduction. The added bonuses are thet if someone is attacking you there is a chance that they will be slowed, and if you activate the item you slow enemies in your area by a minimum of 35% on attack and movement speed.
The Giant's Belt is just for the extra survivability health gives you and just to have an item that builds towards Warmog's Armor
Atma's Impaler not only gives you armor and critical chance, but the passive is that 2% of your health is converted to attack damage.
Last but certainly not least is Warmog's Armor. Starting at 920 health and 30 health per 5, this item gains stacks with kills of minions or champions and they DO NOT GO AWAY. The stats for this item max at 1370 health and 45 health per 5.
This is an expensive build, but devastating to even a fully built enemy team.

I start with blade surge to farm easily. If you get the kill with Bladesurge you get some mana back and the cooldown resets. Using this ability to chase by using it on minions that are low to close the gap can get you numerous kills. Next I put two points in Equilibrium strike. The slow/stun and damage are what will get you kills in the early game. Next I take Hitten style. If you are having trouble staying in lane you can take hitten style before one of the equilibrium strikes since Hitten style give you health every time you attack something and when you activate it it does a certain amount of true damage in addition to your regular attack. The I take another point in Equilibrium strike. Then my ultimate. The ultimate is great for chasing. Not only does it heal you if you hit something with it, but you get four (4) blades to use and you don't have to stop running to throw the blade so this really helps catch the champion you're chasing since that's what Irelia does best. The ulti is also great for minion wave farming, just put yourself in a position where you can send all the blades through as many of them as possible and just fire away.(Note:this is best done when you have low health since the blades health you for 15%of the damage done to minions and 25% of the damage done to champions) The I put another point in Equilibrum strike to try and get the longest duration of the slow/stun as early as possible. Bladesurge is a good distance closer, but the level one has the longest cooldown and least damage so at level 8 I put my second point in it. Another point in Equilibrium strike will max it. Another point in blade surge at 10 makes you a better farmer and better assassin. Level 11 brings the second point in your ulti. At this point you should have part of your trinity force built and at least 3 kills. Although I have played a game where I got 20 assists before I got a single kill. Another point in blade surge to reduce the cooldown and increase the damage. At level 13 is when I take my second point in Hitten style. This ability becomes crucial to your success late game since the active does true damage which just rips through the enemy defenses. Max Bladesurge at 15, and then throw the last point into the ulti at 16. then max hitten style and by now you should be a killing machine if you're doing your job.

The new masteries are up above. This is what I use and I am always open to suggestions on changes or recommendations.

It's a dominion gameplay and I do fairly well.

http://www.youtube.com/embed/kzt53cgfmIs

More will be added to this guide as I am able to complete sections. Please try the build and leave me feedback. Thanks. I will also try and get some gameplay videos over the next couple weeks or so. I apologize for not getting the gameplay, but school has taken over my life. This build still holds true though.