Hey there! I wouldn't call myself a Diana main, but I have been using her quite often more recently in the jungle.
Runes I take magic pen marks, armor seals, MR/level glyphs, and AP quints. Standard fare really. 21/9/0 masteries, taking all relevant offensive masteries for AP builds and getting the jungle defenses from the 9 in that tree.
Itemization-wise, you can actually use her a bit better in the jungle than in lane IMO, because you can get both 10% CDR and 80 AP for only 1500 gold with the Magus enchant, so by the time a mid laner will have half of their first item done you're already sitting on a significant amount of two very useful stats on her. Sorc Boots next. After that, since you've already got a healthy chunk of AP, go for a Lich Bane. Depending on whether you're getting focused hard build either a Zhonya's or, if you aren't, go for that Rabadon's Deathcap. Then the other. Then you're going to want your last item to be somewhat based on the match. This is probably the downside of Diana Jungle when you're playing her as a burst assassin, as you NEED the Lich Bane for burst, the Zhonya's to buy yourself some time when diving a team, the Rabadon's for MOAR BURSTZ, and the Sorc Boots for obvious reasons, which only really leaves you one item to buy. If they've got a heavy AP team, an Abyssal Scepter works great. If they're starting to fear you, which they should, a Guardian Angel here can be really helpful. If they've got a button of tanks, or are just itemizing heavy into MR, build a Deathfire Grasp.
The reason I don't suggest building an early DFG or Zhonya's on her when playing her as a jungling assassin is that, ideally, you're only arriving to lanes when they're ripe for a gank, and you shouldn't need to worry about having to fight one massively powerful target as you're outnumbering the enemy, meaning that you don't need to deal as much damage yourself when you have an ally nearby helping you out, and you shouldn't be needing the Zhonya's for the first 20 minutes of a match because you aren't going to be in many situations where the extra few seconds is needed, like you will later on in teamfights.
The other route you can take with her is as a slightly more fighter oriented build. Again, go for the Magus enchant. Build boots based on the match, if neither Ninja Tabi nor Merc Treads are necessary you can just default back to Sorc Boots. After that build a Nashor's Tooth, so you're giving up on burst damage in favor of sustained damage on her autos, which will be coming much quicker, meaning more passive procs, while the extra CDR from the Nashor's is letting your shield and Q come up quicker. You won't be dealing as much mass damage at once, but you'll be able to cast more often. After those three, the rest of your build should be a mix of survivability and offense. Rylai's Crystal Scepter, Zhonya's Hourglass, Abyssal Scepter, items like those. If you absolutely just NEED the tankiness, a Randuin's Omen or a Banshee's Veil are never a bad thing.
When you're jungling with her, you used to take W first, then max Q, then W, then E, ult at 6/11/16, but because this new jungle hits so much harder that just doesn't work anymore. What I've found that works is going W > Q > W > E and then if you're having a rough match, put another point in W. If not, start maxing Q. Her shield, once you've got an extra point or two in it, actually gives her a LOT of sustain and clear potential in the jungle. I've found that I can actually keep going for quite a long time with her, longer than even my more dedicated junglers like Volibear or Nocturne, because that shield will absorb so much damage while putting out some decent damage of it's own, and then refreshing. If you don't need more than two points, just go ahead and start maxing Q instead. This is going to force you to be a bit more careful about your Qs when ganking, as the cooldown won't be as low as you might like, but that isn't necessarily a bad thing. When you're ganking you aren't worried about dealing enough damage to kill someone on your own, like I said before, so a Q R W E R while working with your laner is usually more than enough to get the job done. If you're careful you shouldn't need more than one Q per gank. As an added benefit, putting more ranks in her W early actually significantly increases her survivability during ganks, so even if the enemy laner tries to attack you when you gank you're just absorbing it all into your shield while your laner goes to town on them and you join in on the fun.
Early on you're going to want to avoid counterjungling, but be cautious if they come into yours. Again, this is where the extra points in her W early make all the difference. It'll actually let you win trades if someone comes into your jungle provided that they aren't pouncing on you when you're fighting a buff, because it's easy for someone to underestimate just how much damage that shield soaks up while it's putting out a solid chunk of it's own against them before refreshing. You can eat at least one or two spells' worth of damage while dealing your own back practically unhindered. If an early jungle fight isn't going your way though, don't be afraid to back off. Even if they put you a little behind early on, your clear speeds will get insane once you have either a Lich Bane or a Nashor's Tooth, so it's easy to catch up through mid game powerfarming or ganking really heavily, so you're getting fed off kills or assists instead. Once you're sufficiently fed though, counterjungling because much easier because after you've got that Lich Bane (Or Nashor's with some durability elsewhere) you're probably not going to be losing many 1v1s, although the Lich Bane Diana is a bit better here. You can Q over a wall to check to see if enemy camps are still up, and if the enemy jungler is there you can jump straight to them with your ult. If not, just hop over the wall, pop your shield to take down the camp, and then go on about your merry way. Try not to focus too much on counterjungling with the aim of finding the enemy jungler though, because Diana really does excel far more at punishing overextended lanes and weak, fleeing enemies. As a jungler she can tower dive EXTREMELY well, particularly if coming in from the enemy's jungle and heading back towards her people. Once you get really good with her you can grab the kill on a low health ADC, pop your shield, and just keep right on walking while letting your shield eat the two or three turret shots, ulting to a minion out of turret range if possible.
Diana is a surprisingly survivable champ, whether through investing in resistances or by relying on sheer AP combined with her shield. You HAVE to be good with skillshots, and you HAVE to know just what your capabilities are at all points during a match so you aren't overcommitting yourself which is all too easy to do. Once you've got her figured out, and have gotten a feel for what kind of situations mean a double kill versus what mean coughing up 300 gold you'll be all good to go.
Provided your laners aren't whiny, needy, abusive, flaming feeders but that just can't be helped sometimes.
Good luck, and happy ulting.