This build is basically for the build itself. Follow it, and you'll do well. I got a 32/0/19 with this build. I highly encourage you to use it.

I covered what I was going to say in the intro up there^
So I will just put Irelia stuff down here:
The Story of Irelia
The Ionians have developed some of the most breathtaking and deadly martial arts on all of Runeterra – just one manifestation of their pursuit of enlightenment. The most remarkable blade style to emerge, however, was the unusual byproduct of foreign intervention. Master Lito was a swordsman whose teachings were sought by ruling classes from nearly every city-state. His art was a highly-guarded secret, but it was said that swords would breathe in his grasp. He withered unexpectedly from a mysterious disease which baffled the brightest of Runeterran physicians. When he died, he left behind Zelos and Irelia, his son and daughter, and a truly unique weapon. Zelos became a Sergeant in the Ionian military and left to seek assistance from Demacia immediately prior to Noxus' invasion of Ionia. Irelia, charged with the protection of their home until Zelos returned, was alone when Noxian forces struck.
The Ionians fought admirably, but soon Ionian blood stained the land beneath the prints of foreign boots. At the Great Stand of the Placidium, Ionians prepared for surrender, but were inspired to maintain their resistance when the young Irelia hoisted her father’s enormous blade and pledged to hold until her brother returned. In the chaos of the ensuing fight, Irelia was cursed with dark Noxian necromancy. As her life ebbed, Soraka, the Starchild, made a final attempt to anchor Irelia’s fading soul. Unwilling to relinquish her home, Irelia rose at the brink of death, and her father’s sword lifted in the air alongside her. Irelia rushed back to the fore, unfazed by the blade's sudden animation. The weapon danced around her effortlessly, cutting down Noxians as they gaped in horror. The decimated invaders were forced to retreat from the Placidium. Irelia was appointed Ionia's Captain of the Guard, and when the defense of her homeland moved to the Fields of Justice, so did she.

"The sword flourishes, as though painting with blood." -- Taken from a Noxian Field Report
Some cool Trivia
Hiten Style is a reference to Rurouni Kenshin, where the main character's martial arts style was called: Hiten Mitsurugi-Ryu. Often known (in the manga/anime) for its blindingly fast speed.
This is reinforced by Bladesurge's ability to be used repeatedly dashing from target to target as they are killed with a blindingly fast speed.
Irelia's swords stab her as she dies. Whether it's out of honor or losing control of her blades, is up to debate.
This could also be an homage to the Demon Hunter hero from Warcraft III (which DotA and League of Legends are based off of); when the Demon Hunter would die, he would fall to his knees and stab himself in the chest to contain his power, making it so that no other would steal it from him in death.
Irelia's telekinetic swords may also be based off of Erza, a character from the anime/manga Fairy Tail.
It could also be attributed to Nu-13, Mu-12, and Lambda-11 from the BlazBlue video game series who all use flying blades as their main attack.
Her French jokes refer to the character of Brice de Nice from the French film named after him.
Irelia was one of the champions chosen for the Ionian pool available during the 2010 Ionian Versus Noxus Grudge Match. However, she was not one of the selected champions.
Skins
The Aviator skin is a reference to Amelia Earhart, the first woman to fly solo across the Atlantic Ocean.

The good runes for Irelia. AtkSpd/Armor, which improve her much.

The masteries for jungling, they work well.

I have discovered that if Irelia has Lifesteal and Strength, she is unstoppable. All that you need is just some other things to boost her up bits.

This is the smartest way to sequence your skills when you are Jungling. It works the best.

Smite for Jungling.
Ghost for fleeing/chasing.

Like I said, all you really need is the build, all this, eh, not so much.