A few months ago, you would have gotten yelled at if you were building him as a tank. He's better as a tank right now, but with the change to Athene's, he can actually go AP again. The damage reduction on his Q to minions is really what's holding him back from being an extremely viable AP champ. His AP scaling isn't terrible and he has CC and mobility with a lot of magic damage. The reason AP Grag used to work so well is that he could stay back and farm all day. If you got even remotely close to him, he could slow you, stun you, ult you away, or simply E away from you. And then he would go back to farming. Now AP Gragas can't just Q to clear an entire minion wave, so he has to walk into harms way to CS. AP Gragas can work, but you have to be so much more careful.
Build-wise: you have options.
You will start either
or
as a first item. RoA will be a more tanky version of an AP build, but you can still do decent damage.
In the RoA build, you will get
the reason you go Lucidity is that CDR is very important on Gragas's damage output as well as his survival. At near max CDR, Gragas's W is on less than a 2.5 second CD which means he can ALWAYS take reduced damage (this is also why tank Gragas works well). I also take CDR/level Glyphs when playing AP Gragas to help get close to the cap. Other options are to go
but grab CDR in another item such as
or
but that will end up doing less for your damage as you will be sacrificing a slot for the CDR. The rest of your build will go into increasing your burst damage in your full combo, so grab
and
to finish out your build.
The Athene's build is all about maximizing damage from your AOE abilities. While using your other abilities more cautiously as you are much squishier. You will still want to get a Deathcap as soon as possible to make it easier to clear waves but you will most likely not build a Lichbane. The full build will look like a standard midlane mage build.
and the last item is a choice to build a defensive item or to get more damage with any of these items
as they all work fairly well with your kit in different situations. Rylai's helps you kite better. Liandry's is okay in siege scenarios and extra pen is always appreciated. And Lichbane adds more burst.
The new item
might be a good second item as it gives you the extra damage you need to clear waves and can boost your harass on a lane opponent. I would recommend trying it out, but only as a second or third item, as it's extra boost in damage is best in the early game. Consider building it as a second item instead of Deathcap, and then save Deathcap til later. I think it has good synergy with his kit as he can cast his Q, let the barrel charge while walking around and using your W to charge your Luden's and then pop it once the stacks hit 100 to get the extra burst in damage. His E gives him a Dash and a cast to help charge his Luden's. Try it out. Peace.