I've been playing League for only a little while, but one of my first purchases was Irelia, and I've been playing her since I was level 13 (I've been at 30 for a month now). Since then, my build has changed drastically, and I believe I've mastered a successful strategy for taking away the late game with Irelia.

What this build comes down to, is an incredibly fast chase and DPS way to play Irelia, perfect for making sure your team gets the Ace before you hit Baron Nashor and/or go for the Nexus.

In the Summoner Spotlight for Irelia, Phreak gave the recommended rune build as full attack speed marks, glyphs, and quintessences, with flat armor seals. This is pretty spot-on, and gets the job done with Irelia.

• 9 Greater Mark of Attack Speed: Attack Speed is important, because her W ability, Hiten Style, deals true damage and restores health per hit, no matter the target; as long as she's making contact, she's going to get the health.
  
• 9 Greater Glyph of Attack Speed: While not as substantial as the stronger Greater Mark of Attack Speed, these glyphs provide an additional bonus to your attack speed.
  
• 2 Greater Quintessence of Movement Speed: For a total of +3.0% Movement Speed, these Quints are wonderful in this build. Apart from this, almost all of the items that Irelia has increase movement speed.
  
• 1 Greater Quintessence of Attack Speed: The final bout of attack speed for Irelia. An additional 3.4% to the attack speed total, bringing the final sum up to 28% additional attack speed. Not bad at all.
  
• 5 Greater Seal of Defense, and 4 Greater Seal of Armor, for some early and late game armor. One of the noticeable *****s in this build is the lack of defense. The Force of Nature and Mercury's Treads I recommend provide some Magic Resistance, and the seals pick up a bit on the armor.

As far as the Masteries go, the Attack Speed, Attack Damage, and Utility build is how it goes. The first tier of Utility masteries is sort of up for grabs; I prefer a little bit of both "reduce time dead" and "mana regen," which will allow you to have an easier time in the laning phase of the game. The only required Utility mastery is movement speed, which really begins to stack up with the runes. The Summoner's Wrath mastery simply gives buff to the two spells I prefer.

If you didn't already figure it out; speed is the way to go with Irelia, movement and attack.

• Ghost will allow you to either get back to lane from the spawn, chase down, escape, or toy with the enemy. When I get to the tactics session, I'll go into more depth.
  
• Surge is awesome; it's my favorite spell to use with Irelia, and it can make her devastating. With the Summoner's Wrath mastery, the 40% Attack Speed increase is remarkable, and clears away the late game. The ability power boost can help too, with Equilibrium Strike and Transcendent Blades.

I prefer to play Irelia with a partner in lane, but I also love to solo top. From the start, pick up Regrowth Pendant, and head to your lane. First skill is Equilibrium Strike, which deals damage and slows the enemy if you have a higher percent health than the target, but stuns the target if you have a lower percent health. If you can lure an enemy champion near your turret and stun them, it hits them HARD.

Pick up Blade Surge (I like to call it Bladerush, personally) at level 2. Use this sparingly though; the cooldown is long, and if you don't kill the target, it becomes very costly. The best part about this skill is that if the target does die, Irelia is refunded about half the mana cost, and the cooldown is refreshed. This can make it an incredible chasing tool if there are several enemy minions nearby with low health. I use this to guarantee the last hit on a minions, and to rush at an enemy champion. A useful strategy early game is to target a weak enemy minion near a champion, hit the champion with Equilibrium Strike, and use Blade Surge again to dash away without suffering too much damage. It's the best way to harass enemy champions, with Irelia's lack of early game range.

Hiten Style is an awesome skill to have, but it's not exactly necessary early game, unless you're taking a ton of damage at the hands of enemy champions. While inactive, you restore a bit of health on each physical attack, but when active, it deals bonus true damage to any target, including turrets. This skill gets much more useful when you build up attack speed as the game goes on, but it helps early game to pick up health and stay in lane.
Don't use recall unless you desperately need to: part of the shtick of Irelia is that she can bounce back from low health incredibly fast if you're careful. Smack a few minions with her basic attack to get life from Hiten Style, just tread lightly near the enemy minions.

Pick up Philosopher's Stone and Avarice blade as soon as you can without leaving lane in crucial moments. Sometimes, I'll wait until I've hit level 6, and have enough gold to pick up Philosopher's Stone, Avarice Blade, and Mercury's Treads by the first time I go back. Building up to Trinity Force takes the longest amount of time, and it's very difficult. You don't really feel the benefit of having it until it's built, apart from the small bonuses from its components. I pick Sheen first, because it's the best way to get a burst of damage from her abilities. If you're being dominated in lane, Phage is a better first choice because of the health boost, but Zeal should always be bought last; the runes will give you enough attack and movement speed to last you until Trinity Force is complete.

Now that you've got your Trinity Force, the real foundation for Irelia, you can start having some fun tormenting your opponents. Laning should have ended by now, and you should have some assists, maybe some kills, and probably a few deaths. Pick up the Regrowth Pendants as soon as you can, and stay in lane as much as you are able.
I want to talk about Irelia's ultimate now, Transcendent Blades. In my opinion, this is the black sheep of her skills. It's a ranged attack that shoots four separate blades one a time when you hit the R key, in the direction your cursor is facing. You can be moving or attacking or using a skill when you launch them, which makes it incredibly versatile. Upon hit, you restore 25% of the damage done to enemy champions, and 10% of the damage done to minions or monsters. I use this skill for three purposes:

1. To shear away minion waves when approaching a turret. This can clear away a minion wave, which will allow your own minions to keep moving down lane until they become turret fodder. It might seem like a waste, but it's fast, and can really help pushing when enemy champions are elsewhere.

2. To chase down enemy champions. I feel like this was the original intent of the skill, given that Irelia's mainly a chase champion. You're running after a champion, and you launch it at them to deal some damage, plain and simple.

3. To regain some serious health in-lane. This skill can help Irelia stay in lane remarkably well. Shoot down some minions, or hit some champions from a ways off, and you'll be back up to snuff in no-time. I've picked up half my health bar by using this skill in the right situation. Played well, you can turn the tide of a lane with this.

Now then, back to the strategy. As you get into team fights, Blade Surge into whoever your team is focusing, but save Equilibrium Strike until you can stun or slow a fleeing champion, or a champion focusing one of your allies. A well timed Equilibrium Strike can be the difference between life and death for you or your teammates. When you grab your Force of Nature for just another 1740 gold, the fun really begins...

I'll be writing this in detail soon. In the meantime, I'll give a quick summary to finish up:

Now that you have your Force of Nature, be sure to tease your opponents to no end. You'll likely be faster than the entire enemy team, and with Ghost and well timed Blade Surges, you can escape from almost any situation. Ionian Fervor (Irelia's previously unmentioned, but incredibly useful passive) stacked with Mercury's Treads' tenacity will prevent you from being held down too long; but be careful around teams of 3 or more. When you pick up Youmuu's Ghostblade, you're a certified death machine.

Pick up Phantom Dancer to simply enhance the Attack and Movement Speed Irelia already has, in addition to bolstering the Critical Strike Chance, which should be pretty high by now as well. Finally, you can sell the Philosopher's Stone for any item you want. I prefer the Madred's items because they allow Irelia to take more of a hit than she previously could, as well as hit a little harder and faster. The passive on Bloodrazor's isn't a bad addition either, but by this point it's not necessary, it's just a slight, welcome buff. Other options include a Bloodthirster, for a more damaging option, or even an Infinity Edge, if you prefer a higher Critical Strike path.

More tactics to come, but in the meantime, enjoy the build!