OK, I've watched a lot of AP Zac videos and I notice a lot of people play him a lot of different ways. So here's how I play him with the most success/fun.
*Max his Slingshot first. Slingshot has the most damage (even more initial dmg than ult), is your go-to CC ability, and gains range and reduces cooldown with level. Even in lane this is what I do. I see lots of people lvling his Stretching Strike first, but that's the one I do LAST. I want to jump in out of nowhere, then Ult while spamming my W.
*Don't bother with boots. You might think this is crazy, but Zac (especially AP Zac) relies most on Surprise, CC, and a huge burst of dmg, in that order. 90% of my fights are done in some form or another in just a few seconds. But if you want boots, then sure, go for it. Not a HUGE deal.
*My build order usually goes like this:
-Rabadon's Deathcap
-Abyssal Scepter
-Luden's Echo
... after that I don't think it matters too much. Hourglass. Liandry's, Rylais, WOA, Void Staff... whatever you need. Though I will mention that a little bit of spell vamp goes a long ways in a sustained fight.
I've attached the results from some of my most recent games with AP Zac, most of them Ranked Games at Gold Level. And I will note that even when I lose, I pretty much still have an awesome Kill/Death ratio. (I know I haven't played for 2 weeks, but I'm trying to cut back while I do some writing. I know, real life stuff right?)