Small Golems (3 HP Pots, NO SMITE)
Wraiths (smite big one, HP pot)
Wolves (No smite)
GANK -> b -> buy razors & HP pot
Blue buff (smite & remaining HP pots, if you feel safe smite early to allow fast cooldown)
Red Buff (smite)
Jungle route finished. Once you have lv3 hiten style jungling is a peice of cake so feel free to do what you want from now on. Gank as much as humanely possible.
I'm not going to tell you how to play irelia... you should already know that. This build is there to tell you the route and an item build that kicks *** for this irelia. However I will explain my maxing hiten style first among other things.
Maxing your E on irelia is better for pretty much anything except doing a massive amount of damage in a very short time, at LOW level. My maxing your hiten first you will have an immense amount of dps for a level 9. Enough that your ganking and jungling will be VERY fast and VERY strong. This will then catapult you up to 13 for your maxed E. Q is now pointless to max since the patch, you have to rely much more on your auto-attacks for damage than before, you can't just q + trinity proc + e for 1k anymore then sit back with a few auto-attacks and ult them into the ground. Your q hits much less hard and your ult doesn't heal enough to keep you alive in a fight (although it does hit harder).
By maxing hiten style first, you will out-dps yi/tyrnd/ashe/vayne/corki you name it.
Irelia can no longer go tanky anymore very successfully as she doesn't heal nearly as much and her nuke is significantly nerfed. Your going to have to get in there and kill em a lot more.
Madreds rushed after boots will do this for you, it will make you do freakishly high amounts of DPS with your W on, and the armour combined with the AS heal will keep you alive. Next it's sheen & zeal for the extra burst, speed and more AS. I then get emblem, your doing more damage so a bit of lifesteal will go a long way to keeping you alive before maxing trinity.
That is your core finished. High speed, decent survivability and a LOT of damage. Next I tend yo complete my starks which will take my damage to as high as it will never be needed to be.
After this banshees will give you a very good boost in survivability, the last item however is completely down to free choice and the enemy team, but I would suggest a tanky item, examples:
Sunfire Cape (against the DPS teams)
Force of Nature (against the caster teams)
Aegis of Legion (if no-one else on your team has it, GET IT)
Hexatech Gunblade (will make your ult NUKE HARD and lots of health regen for survival)
Malady (if you have a lot of casters this is nice to get)
Runes: You can manage (just) without the quints but you NEED the AS marks AND armour seals for this to work. Glyphs are personal choice.