A lot of people are wondering how Leona jungle looks in season 5. The clear speed took a hit with the removal of Madreds, however you can still get to level 3 and gank often. She can clear camps just fine when her smite is up, but her lack of sustain makes it difficult to clear otherwise. If you save smite for wraiths then it should be PLAYABLE, but i'm not sure if it is as viable as before.
Today i'm here to propose an unconventional jungler that has always been considered as a possible pick, but never really seen any real play other than 'troll games.' I'm going to try to convince you that Leona jungle is very viable and hopefully you will try it out at least once, if not just for fun. I've played this against various players to great success, and I think i've finally found the 'recipe to success' with Leona jungle.
First lets establish: why would i pick Leona jungle?
Leona is among the highest cc champions in the game, so naturally her kit would be great in the jungle. She brings a root and point and click stun as early as level 2, and at level 6 she can provides upwards of 3 seconds of chain lockdown. If i could compare her to anyone, she is very similar in function to Nautilus, however she is stronger in many ways. Her E travels through minions, and her ultimate is much easier to initiate with and is on an extremely short cooldown.
You pick Leona for her overwhelming crowd control.

The reason i pick this rune setup is mainly to fix Leona's 'bad clear time.' People use this as an argument against picking Leona or even Nautilus, but with these runes and masteries it is improved significantly. Do NOT get tricked into running AP runes because of her ability scalings. Not only are they lackluster, you have to consider that her overwhelming CC means she gets a lot of free auto attacks while the enemy is stunned. The damage stacks up when you go AD.

This is where things get a little interesting. Get as fast as you can stalker blade, to help your jungle, and also have another CC to your ganks .
After Stalker, you want to rush boots of mobility. As Leona, you don't want to be farming the jungle too often, instead you want to gank CONSTANTLY due to your extremely effective ganks.

You farm in between ganks

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After that, you build tank according to the game. Sometimes you'll want armor, sometimes you'll want MR. You do not need any damage; you will notice that you actually do a respectable amount of damage already. The best single damage item is Sheen. This gives her a noticeable damage boost if you weave in sheen stacks with your abilities. As such, Iceborn Gauntlet or Trinity Force can be good items.

I prefer to max E for the lower cooldown. This will enable you to get multiple gapcloses in an extended skirmish or fight. A good argument can also be made for maxing Q or W. Maxing Q will give you maximum damage output, maxing E gives you the most mobility, while maxing W will give you the most tankyness. Choose according to your playstyle.
Tip: You can get three auto attacks off in very rapid succession by using auto > Q > auto. This is essential for clearing the jungle fastest.

Do your first buff and immediately level 2 gank mid. If you land an E, it will guarantee your Q stun and it will force the enemy mid laner to burn flash or die. Mid laners will NOT expect you to level 2 gank so do it every time! You do not need to start red for this to work, but it works best with red buff as it will apply a constant slow after your initial crowd control.
Tip: Sharing two minions with your mid laner will enable you to do your first buff, gank, and then second buff and still hit level 3. You won't need to clear a small camp inbetween buffs.
After that, look to gank as much as possible. Your ganks are disgusting because of how good your engage is. With mobility boots you can just run around ganking, and then clearing camps in between to maximize your time efficiency.
At level 6 is when things get fun. A 1200 range stun is extremely broken; use it whenever it's up in a lane. The enemy will be forced to flash or die, as landing your ultimate makes it extremely easy to chain with your E and Q. It also has an exceptionally low cooldown

Late game you transition to a hard engage tank.