Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This is the build I use to play Irelia as a tanky DPS jungler with a very fast clear of the jungle, great burst, great sustained and hard to kill champion that is very strong in the current metagame. Your basic play style should take it easy early game, till 5-6 you can gank and burst a large amount of someone's health, late game revolve around a hard to cc off-tank/AD carry.
Very hard to CC
Fast jungler with decent ganking power
Mid/Late game very tanky and very hard to shut down and kill
Squishy early game
Low DPS early game
Very expensive (but easy to farm with)
Variation on the build
There are many variations to the build and you should always adapt it to the situation. Youmuu's is exchangable with Wit's End for your early item, or even Sword of the Divine. Wriggles is not always needed, and what tanky items you build should vary each game.
Masteries - 0/22/8 and start with Doran's Shield is good. 21/9/0 or full defense is best for the laning Irelia.
Runes - Full AS runes are the best for Irelia and will give you a very fast clear of the jungle, but these are expensive and undesirable runes for most champions and thus ARP reds, Armour yellows, CDR blues and HP quints are also viable.
Skill sequence - Leave it alone, Bladesurge is pointless to level beyond 1 early, Hiten Style makes your jungle fast and sustained, gives you staying power and good burst especially vs armour heavy characters. If you want to lane it might be better to level Equilibrium Strike first.
WORK IN PROGRESS
This will be update later when I have the time with explaination of skills, summoners, item order, ranked play, and in depth analysis of how to play Irelia, but really the guide by Heat covers these really well, I just dislike the direction of item build and find it archaic for the current meta.