From what I can tell, much of your issue is in your build.
She's very bursty, and even at low levels the ability to Q R W E R is enough to kill your squishy ap mid enemy laner when you get a Lich. While normally a Lich is a later item, since you get more burst from it, because she can so consistently proc the damage from her combo it works great on her as your first buy.
Diana during the mid and late game probably shouldn't be having many mana issues, and you shouldn't need MR early either as your objective isn't to trade, just to farm for six. She's like Akali, you hit six first and you're pretty likely to win the lane. Don't worry about the Seraph line or a Grail, because she just plain doesn't need them.
After your Lich get Sorc Boots, a Zhonya's hourglass. This will allow you to safely dive into teamfights during the mid and late game, pop your Q R W E R then Zhonya's to buy yourself some time to get your Q W E back off cooldown, and to let your team capitalize on the fact that you either did heavy damage to, or completely took out, their carry. Then get a Rabadon's for obvious reasons. By this point your Q should be enough to clear an entire wave, and every third attack you'll be sapping a solid amount of health from anyone you're fighting. If their ap carry is starting to become an issue, get an Abyssal Scepter. It also comes with a neat passive that'll further reduce their MR, which your entire team benefits from. For your last item, consider getting either a GA, if you want to play it safe, or if you want to be a bit ballsy you can get a Nashor's Tooth. That'll let you duel anyone in the late game if your burst doesn't completely destroy them.
Learn how to tell when it's safe for you to go in. Whenever possible try and get two stacks on her passive before diving someone so after you go in on them, you have that bonus damage on your next auto. It'll greatly help your burst damage. Use her mobility to your advantage, darting in and out of fights, using your shield to stay alive long enough to lay down a beat down, and her knockup to disrupt during teamfights. A solidly built late game Diana can burst down most enemies on an enemy team and push down lanes like nobody's business.