Diana is a great jungler but takes some takes some time to get used to. With her your role is to close gaps and not really to do any real damage. Your positioning is key to playing Diana do to the fact that all of her moves roam around her Q Cresent Strike.

In the jungle you will want to start at at Wolves then go to Ancient Golem because she is somewhat mana hungry at the beginning, then go to Wraiths, back to wolves (mini golems hit hard), then the Lizard, and last mini golems if you can. If you got a good gank at blue by the time this path is over you will still have almost all of your health and you get to roam/gank or you can go back to fountain and go on a shopping spree.

Now in the jungle you want to start off with her W(Pale Cascade), this give you some sustain in the jungle as it is a shield and it helps you out a lot in the beginning, Next you want to get your Q(Crescent Strike) which deals damage and helps you clear a little bit faster, Next lvl up your W again to help you with that sustain, at lvl 4 get your E(Moonfall) which isnt any help in the jungle but is awesome with ganks. Now just Max your W first then your Q then E last, and obviously lvl your ult when its available.

Now when ganking with Diana shes really not useful until lvl 6 when she gets her ult but at early lvl's try to get as close as you can to the enemy use your W for the shield and if they start getting away use your E to pull them back in (that should be a kill). Now when you have your ult you will use Crescent Strike and then ult on on them, then pop your W and again if they are getting away use your E, if they flashed out and are weak ult on them again and get the kill. The main thing about using her ult is that you want to hit that first Crescent Strike then ult cause that will reset your ult with no cooldown so you can use it a second time if they escape.