I always look to counter jungle at level 1 if my team mates are down for an invade leash.
Head for the enemy blue side and ward right between the blue/gromp where he should walk in.
I'm sure you're aware but if you ward more than halfway over the wall it'll drop to the other side, that way you don't walk too far and risk getting caught in the face with cc.
If you don't spot anyone it's safe to assume he started red side. If your team mate is down to invade leash, you're free to do his gromp/blue/then skuttlecrab. Back. Your krugs/red/gromp/blue.
From there adjust your route in accordance to what's happening on the map. Ex: Your top lane is pushed up far and they have someone like lee sin who is great at early ganks, chances are he will be looking to gank so head for your top lane and look to countergank.
Now if you do spot the enemy with your early ward you have a few options.
Since he is starting blue side, make a dash for his red side if your team mates are down for the invade leash. Have them ward a common path just in case someone is in the area. Make sure you take a path where the enemy doesn't see you moving towards their jungle.
Start their krugs/red/skuttlecrab. Back. Your Gromp/blue/krugs/red.
Again, adjust your route depending on what's happening on the map.
Your team isn't down to invade leash the red side. Now I'm sure the common choice is to get a leash on your gromp then do blue and maybe skuttle before you back, but I would start red side and go krugs/red/skuttle. Back. Then hit your blue side and adjust your route in accordance to how the game is playing out. My reasoning behind this is most people are gonna expect Diana to start blue side, so if you're up against someone who LOOOOVES the level 2 cheese invade like Shaco, Nidalee, and Eve they're gonna expect to catch you doing your blue buff but AHA you aren't there. Also, I feel like dianna would benefit from starting red side because her first few clears are pretty aa dependent and the red side is better for that.
Well you've spotted him on his blue side so you know he'll be starting there. GO FOR THE CHEESE. Catch him doing his blue buff at low health and go for the eassssy kill. MAKE SURE YOU GET SOMETHING IF YOU DO THIS. If he gets away somehow, make sure you blow his flash and/or snipe away that blue buff that he was so kind to weaken for you. You can reallllllly mess up someone's early jungling if you get that cheese kill, it'll set him behind in gold and exp. Take his wolves while you're at it if the enemy laners don't go mia.
From there, just farm your camps (and his) as quickly as possible. Get that 6 and get some ganks off. If the enemy is playing smart and you don't have a guaranteed kill somewhere just farm farm farm farm farm. Don't try to force a gank that isn't really there just because you're 6 now.
One HUUUUUGE thing you can do to stay one step ahead of their jg is to pink his blue/red and drop some normal wards on the other side.
If you spot him in his red side, go take his gromp/blue. Or gank the opposite side if the map.
Just make sure you are farming no matter what happens even if you're playing like a boss and getting off awesome ganks, dianna may be a great ganker post 6 but kills/assists alone aren't going to keep you relevant forever.
Especially because you want to get devourer and start stacking it ASAP, make it a personal goal to hit sated by 20 minutes every game.
I am in no way shape or form a Dianna expert, this is just my opinion on the subject and some things I've started doing with every jungler recently and I've seen some good progress in my jungling. It used to be my worst role because i didn't even really think about the enemy jungler and consider what he was doing, I've seen a lot of progress in my jungling after i started doing this and it has quickly become my favorite role because of the amount of impact you can have on the game if you're doing it right.
I've really gotten into the jungling mind games and now i have a blast playing jungle, there's nothing more satisfying than denying the enemy jungler to the point where he's scared to walk into his own jungle and knowing he's getting frustrated because his camps are always gone.
Also, swapping out your trinket for a sweeper on your first back is a great habit to get into.
The key to playing anyone in the jungle is to just straight up dominate the enemy jungler, take over the map. I can not stress how important it is to have vision in his side. With good wards in his jungle, you can predict his every move and be one step ahead at all times. It can put him on tilt pretty easy if you're dominating his jungle.