I bought Irelia right off the bat. She was the second character I was excited about. Lux being the first. When I first started with her, I could not play her for ****. I was squishy and fed like there was no tomorrow. But I was persistent, and after some theory crafting, I went from feeding to being fed.

Masteries
These are basic 21/0/9 masteries, and should be altered to best suit your personal needs.

Summoner Abilities
Ghost is an amazing spell, period. Whether used to catch up or GTFO, it's practical and one of my favorite spells. Irelia isn't all too squishy and hits hard enough to kill before killed. But for those sticky situations when you need to run, Ghost comes in handy.
Exhaust is good when doubled up with Irelia's stun. Slowing the enemy champion down and then stunning them gives you the opportunity to get the kill before they even have a chance to decide to run.
But again, I usually run Fortify because no one ever does, and I like to protect my towers.

Skills
I max out Bladesurge right off the bat. It's almost the same as Shunpo for Kat, but if the target dies, your mana is refunded and the cooldown refreshed. It's a helpful get away tool, but also good at catching up to your target. Afterward, I go back and forth with Equilibrium Strike and Hiten Style. They both have they're uses, but I don't believe one to be more valuable. One stuns and one increases true damage as well as heals you with each attack. And of course, getting Transcendent Blades whenever available.

Runes

Greater Mark of Alacrity - 9
Greater Seal of Resilience -9
Greater Glyph of Alacrity - 9
Greater Quintessence of Fortitude - 3

I stack Irelia with attack speed, armor, and health. Here are my reasonings. Irelia is a hard hitter, and the faster she hits, the more damage she'll do, which means your opponent will die almost instantly from auto attack. Irelia's base armor isn't too amazing, and since she is a hard hitter, it's important that she stays alive. So armor is a necessity. And with health, she'll require more hits to be killed, and with some armor protecting you, you'll last much longer in team fights, and will help your team dominate.

The first thing to get are Boots of Speed and Health Potions. After you've got some money grab your Beserker's Greaves. My first big item is a Guinsoo's Rageblade. At 8 stacks of the Rageblade, attack speed will be pretty decent. The stacks drop off pretty quickly, so it's important to keep them up by slapping minions around. The next item to grab is a Sword of The Occult. Attack Damage is good, don't ever forget that. And with the attack speed runes, you'll attack fast and hard. The Bloodthirster is my next item for several reasons. First off, the attack damage is pretty freaking sweet. And secondly, life steal is a must for any hard hitting melee. Now, my next item I'm a little skeptical about. Nashor's Tooth has good attack speed and helps regenerate what little mana Irelia needs. The ability power and ability cooldown, however I could do without. But with the other stats, it's a good item to throw into the build. My last item is a Phantom Dancer. Attack speed, crit, and movement speed. Enough said.

I'm going to try and throw in as many scenarios in here as I can. Let's start with laneing.
If you're taking a lane with someone, you should be the one to initiate. Bladesurge to get yourself to them. If they decide to run, Equilibrium Strike is your next option. If they do decide to stay for the party, use Equilibrium Strike anyway, for the added damage. Afterward, use Hiten Style to heal yourself and increase your true damage, which will increase your auto attack while you wait for Bladesurge and Equilibrium to come off cooldown. You'll easily dominate. Bladesurge can also be used to escape. Say you're low health, you're close to an enemy turret, and you've just killed your target. Bladesurge to enemy minions that are most likely behind you, combating your minions, and then Ghost away to safety.
I don't jungle with this toon, but I have tried it. I despise jungling as a whole because I don't like being vulnerable to ganks or having to face the humiliation of being massacred by a minion. But if you're going to jungle, get Hiten Style first, because you will heal as you attack, which gives you a better chance of surviving.
Going mid is not something I usually do with this toon, seeing as she's easily combated by armor items, and usually there are better mid champions on the team. But if you are stuck mid, play defensive until you get your ulti. The most you should do to avoid unnecessary death is to auto attack and then quickly retreat a few feet.
Personally, I prefer to lane with someone else.
There are only a few things to remember about Irelia. One is that she is very vesitile and can do almost anything. She can tank, she can dps, she can jungle, she can lane, she can mid. She's an extremely valuable player, because depending on your items and summoner abilities, you can do almost anything.
Another is her ulti. What really got me the first time I played her was her ulti. I was shocked when I hit my R and it only went off once instead of the four blades as the tooltip suggested. It wasn't until after a few seconds I realized that if I hit it again, another blade would launch. Only by then my ulti was up and the cooldown time showed. A lot of people who are new to her don't understand that you have to hit R four times to launch all four blades. And they fire in whatever direction your cursor is pointing, not which way she is facing. So if you're running away and you pop your ulti, remember to put your cursor behind you, or in the direction of your opponent.
Irelia is a tricky champion to master and you're not going to understand her right off the bat. She is rather difficult to get a handle on. But practice makes perfect, and sooner or later, you'll be the fed instead of the feeder.

This is only my second build, so any suggestions are welcome. I hope that anyone who plays Irelia and uses this build pwns face and fully enjoys playing her.