The first thing I am going to say here is that jungling is a team effort. DO NOT expect to succeed at jungling with any character if you have zero help from your team.

Give about 30 seconds to a minute warning before a gank.
Make sure your team knows to call to you if they think a gank is possible.
When ganking, make sure you go in as a team, if you single file, you will all die.
If possible try to give your carry the kill, you will get fed off of assists, they will win the tower and team fights.

That said, pulling off a successful gank is a wonderful feeling.

Diana is a burst ap champion. She excels at killing as soon as she hits level six. She has a bit of built in toughness, a 'You no escape', and a dash that you can use in conjunction with her basic nuke.

*Note: Your ult always takes priority
Q: Crescent Strike
This is your main ability(first priority) you will be using. This is your initiation, your harass, your start of escape, and it allows you to see into the fog of war//bushes. This ability works in conjunction with her ult. If cast before lunar rush, lunar rush's cooldown is immediately refreshed.

W: Pale Cascade
This is your second priority skill. This skill is used as a defense reaction(you know you are about to be hit) or pile up damage versus your enemy. Pale cascade grants you an immediate shield followed by three circling orbs that if hit an enemy strengthen your shield.

E: Moonfall
Moonfall is your early ganking maneuver and anti escape skill. This skill will frequently screw up an enemies flash, so use it if you feel they are trying to escape. Do not use this move to initiate. This skill should be maxed last.

R: Lunar Rush
Lunar Rush is a dash that does damage. This lets you do lots of reckless maneuvers like tower diving. However you should try to hit with your q before using this ability, that way your cooldown will be refreshed. You can use this maneuver in conjunction with your e to prevent people from escaping from your group. This ability is pretty straight forward, but lets you do some fun maneuvers.

If you are trying for an early gank(pre 6), you will need your E. Feel free to pick it up as your third skill point if you feel it will get your team an early kill.

Jungle Route A. (does not need a leash, though still helps)
Blue Buff - Ancient Golem
Wolves
Wraith Camp
Red Buff - Lizard Camp
Small Golems
Wraith Camp
Wolves

Jungle Route B. (needs a leash)
Red Buff
Wraith Camp
Wolves
Blue Buff
Wraith Camp
Small Golems

Route A is a bit safer. Diana is a little mana dependent Much like Amumu, she can get around the jungle with ease, when you have blue.

Route B is a little safer if you are worried about an invade and it puts you at bottom lane at the end of your route. THis will allow for a gank if your bottom lane is having difficulty.

Note: In either case you are dependent on blue buff until you get your first mana item. (Ward the enemy jungle and take theirs so you can give yours to mid)

Void Staff -- Replaces Abysal Sceptor If enemy is stacking a lot of magic resist

Morello's Evil Tome -- Use this item if the enemy team has a lot of healing.

Zhonya's Ring - If you are concerned about dashing in and dying, pick this up and use it instead of a will of the ancients.