Not Updated For Current Season
This is not a build for those inexperienced with Zil. This is a lean solo/carry build that emphasizes massive AP damage and generally being the scariest most frustrating thing on the field. With 19% extra xp a Zil carry is one of the nastiest pieces of work out there.
Flash is a must. It makes Zilean pretty much unkillable. Whether you've popped chronoshift and run against a wall, or just plain flash and time warp, it takes a hard to kill hero and brings it to the next level. In my eyes, no zilean should be played without flash.
Less necessary than flash, it's just what this build uses for that little extra push of damage. The only other skill I would consider using on Zilean would be cleanse if you wanted to lean more towards survivability.
9/0/21 is the only thing that makes sense with this build. Basically maximizing mana/manaregen/damage/cd reduction all at the same time, which is everything Zil needs/uses. Burning embers gives you only 10 extra AP while ignite is down so it's in no way worth giving up presence of the master, which will free up your flash and ignite more often.
Start out with sapphire crystal and two heal pots. This gives you staying power in the lane, and enough mana to harass and get some creep kills. First trip back to base you should pick up tear of the goddess and boots, go into lane and just farm hard. Pick up a Mejais asap. From here, finish boots, finish Archangels staff, and then it goes Zhonyas, Deathfire, and Lichbane. Late game if you have the money and Mejais charges aren't doing great sell mejais and pick up Rylais instead for some bonus Hp/Ap.
Early levels with Zil don't play too aggressive. Harass only when it's easy/ they come into range and only go for kills that leap into your lap. You have enough mana/regen for steady harrasment, but pace yourself, you want to save a good amount of mana for level 5. A few bombs and they should be scared enough that you can concentrate mostly on last hitting.
At level 5 your bombs are really starting to hurt. Rewind+Bomb will take most of any heroes HP especially since you'll be getting levels faster. Level 5 is when you start diving with time warp to set up a double bomb. By level 6, your opponents are not safe behind their towers.
Tower dive skill order: Timewarp+Flash+Bomb+Rewind+Bomb+Chronoshift+Ignite
Then at 7 and 9 respectively you get even bigger damage, try to maximize XP/farming till level 9.
At level 9 pick up Mejais and start visiting other lanes. You are level 9 they are 5, two bombs and if they are missing even a little hp then BOOM. From 9 onwards you will be constantly on the move with Timewarp speeding you along to the next gank, or to catch a tasty wave of mobs that've been building. Level 9 is in essence the height of your potential, everyone on the other team is relatively squishy right now. Go wild, get some charges and farm like mad.
With time warp leveled enough, you will be all over the map all the time, baffling your opponents by showing up where they least expect it. By the time you have Zhonyas, you'll be mostly in team fights; giving you some versatility if your getting focused. It's nice to pop a chronoshift on your Jax, drop a couple of bombs, then Zhonyas and Timewarp/Flash out. Be VERY COOLDOWN AWARE. If you watch the map and your own Cooldowns you will never die. Hang behind your team, buffing and punishing at will. When you pick up deathfire you have a great additional source of damage to nuke with, and the cooldown reduction is big news as well. You will be able to kite with Timewarp indefinitely so you just drop bombs and run back or chase as required. When you pick up lichbane you will have 500smthin AP, basically no cooldown on anything, so each physical hit will deal HUGE damage based on AP, making you that much more badass.
Zilean is basically a hand of god character, 0 auto attack potential(minus lategame with lichbane) so everything is completely up to the player. AP Zil is just as much about support as it is legendary nuking, the trick is blending the two together in a hero with huge survivability potential.