Hello, I am here to introduce one of my favorite champions in one of my favorite roles. As a warning, this will cause a lot of people to dodge games and if your game goes badly you'll probably be blamed for it. But that's kind of what jungling is all about. Despite not having much real CC, I feel that Akali can do quite well in the jungle. Like all Akali builds, it does depend on her getting kills and assists to truly become a monster.

Pros:

Has good sustain in the jungle after completing Revolver
Tons of damage, especially for a jungler
Can shroud to escape after a bad initiate or to save teammates and turn around a gank
Sticks to enemies better than anyone after Rylai's is complete

Cons
Not a very good first clear and is weak at level 1
Takes a lot of kills compared to more traditional junglers
Not the kind of champion to solo dragon with early on (one of my favorite things to do when jungling)
Really needs a Madred's before building any AP

Like most if not all junglers, Akali is going to want to start with a machete and 5 potions. With any kind of leash you should have no problem taking out the blue buff and wolves. She doesn't have the fastest clear so shortly after wraiths it's time to take red. Then she should be level three. It would be good if you had enough to buy Madred's and boots on your first back but it's not necessary. Either way, leave the Madred's as just that and rush a Hextech Revolver immediately afterward for sustain and damage.

I recommend going Spectral Wraith as soon as possible. I know that's inefficient because you already used a machete but trust me, you're gonna want those Madred's procs. Rylai's is very important to this build. After finishing it, anyone who gets near you is basically gonna get dashed to and chain slowed until they die. It's also the first item that will make you tankier. Giant's Belts are always good. After that you can go with any item in the core you think should be next. I don't always build in the same order or even the same things all the time.

Akali's damage is very potent unless you've fallen far behind. Anyone who suddenly sees an Akali show up opposite them in their lane is probably going to be pretty scared. I find flash to be absolutely indispensable for Akali jungle and half of the reason is to secure the close kills early in the game before you get your ult. If you can save your flash by just throwing down Shroud right in the path they're running through, that would be an ideal gank. Basically, play just like a roaming Akali would but ward more.

There is absolutely no finesse to it by the time you finish a few core items. If you've been farming the jungle and taking over lanes for people, you might not even need that many kills to start doing the big damage. Rylai's early on is the most important thing in the world. It basically means no one can ever escape you again. This Akali absolutely excels at diving turrets after the enemy gets pushed out of the lane.

Her counterjungling is pretty good as well. She won't win a straight up trade against every tanky bruiser in the jungle but she should be able to at least escape any bad situation she gets herself into so you can feel pretty safe stealing buffs and warding their jungle.