Zilean is a ranged caster with some heavy burst damage and support utilities. He specializes in harass, insane burst damage, and overall just being extremely annoying. In other words, Zilean is one of the most annoying champions to deal with, and can be played either as a support or as an ap mid. This guide is mainly focused on his ability to go ap mid.

Yes, some people think he isn't a viable ap mid
No, they are not right
Yes, you will prove them wrong
No, you shouldn't ult them

Pros
-Great support utility(slow,speed,revive)
-Heavy burst damage
-Strong in many team compositions
-Great harass in lane
-Cool down reduction
-Has the best joke
Cons
-Squishy(unless you build rod of ages and rylais)
-Extremely mana intensive
-Somewhat hard to play
-Old

The build shown above is mainly to do as much damage as possible by building flat AP. The double is your only source of defense in the laning phase. I normally go this build, leaving me extremely squishy, but that isn't a problem if you know when to ult and when to go in.

Optionally build instead of double doran's, catalyst the protector always first. If you go this build you don't need to build
as you will already get the mana from the

The second build would be something like this:
catalyst the protector

Other items:
kage's lucky pick Good early game, give gold and ap, can be turned into or
If you are really fed, and have enough ap, this could potentially be a really good item on Zilean to burst someone down. The gold per 5 helps a lot if you somehow have trouble with farming.
rylai's NO, if you need health get a rod of ages in the beginning, and you already have to slow people.
Good for laning against somebody else who has burst such as
CDR, mana regen, ap, and magic resist. Really good item for early levels and even later levels too.
NO the magic pen and what little ap it gives isn't worth it at all
NO, you're not you don't have anything that scales off of an auto attack.
It can be good, but you have to get super fed and just isn't worth it imo.
It gives ap, you can turn yourkage's lucky pick into it, but don't let this get in the way from building something important like
A wota can be good if you have someone like top, but I see no point in spell vamp on Zilean
NO
NO
Can stop you from being bursted by someone like

Take all ap masteries and some utility masteries for the extra and movement speed. You can also go
For that extra defense that might just save your life.

Greater Mark of Magic Penetration You always want to take magic pen marks on AP casters. You cant just let them stack MR in the beginning that will make you do no damage.
Greater Quintessence of Ability Power Self explanatory
Greater seal of replenishment Zilean has some serious mana problems, and to counter that you need some mana regen runes. Also, helps too.
Greater Glyph of Magic Resist You don't want to be completely defenseless when you're in lane or else a or will completely destroy you.

NO
Maybe against heavy damage dealers but still I would say NO
NO you have
NO
NEGATIVE
NOPE
NO
You should be getting blue buff 24/7 so... NO
Stops healing, good for finishing off people
NO
YES FLASH OP
YEA OP SUMMONERS get it to jump across the map to save teammates, gank, or save your lane
What? Dominion? NO
What? Aram? YES
YES
YES
NO

Invading
Zilean can help a lot in an early invade. You usually want to get your Time Warp in case your team doesn't have as much cc as you hoped. If not grab Time Bomb to help with damage.

Laning phase
Always last hit with your auto attacks. Zilean's base damage on his auto attack is relatively high, and it makes it easier to last hit. If you're going against an easy lane, you want to harass, but not spam your to the point where you have no mana. If you're going against a harder lane such as you want to avoid the damage and focus on farming.

Ganking
If you decided to bring then you should gank every time you push. You don't even need teleport to gank. You can go through lane, or right through the river if they didn't ward. Don't underestimate Zilean's ganking potential. He can slow and burst you down so fast, you won't even see it coming. However, there are such things as bot lanes that actually ward, so it won't be easy, especially if the other mid calls his mia's correctly.

Team fight's
This is where you shine most. When late game comes around with your full build, (or even part of it) this is where you get to show exactly how annoying Zilean is. However, there is a way to approach team fights.
-NEVER, EVER INITIATE TEAM FIGHTS
You are not the initiator, you have no hard cc or health to even help you survive the team fight. You are welcome to come in during the middle of the team fight to help out, but NEVER initiate. EVER
-Try to go in when the other team used all their CC moves. such as Crescendo or Stranglethorns
-Do not go b3st m3lee chump NA mode as Zilean. You do your damage from a safe distance with your bombs, don't take it for granted
-Always Chronoshift the ADC when s/he gets low. If your ADC is not in danger, use Chronoshift on somebody else who is low. Always ADC first though.
-Spam that Rewind

Skills
-Always Time Bomb then Rewind then Time Bomb for the maximum damage.
-When leaving base, Time Warp then Rewind to save time
- Chronoshift carries and spread bombs apart in team fights

LeBlanc Gay champ, instant burst after level 4, hard matchup
Kassadin Gay matchup, silence screws with your combo, he doesn't have as many mana problems as you do
Ahri Really hard lane again, but as long as you just farm and dodge charms, its not a big deal
Cassiopeia That poison
Cho'Gath Silence, knockup, ult does tons of damage.. what a pain
Diana No. Just no.
Anivia Really want to dodge her Flash Frost or you might get bursted down
Kennen you can dodge his abilities pretty easily, and burst him down, just don't get caught in his stun
Lux Easy lane, dodge all skill shots and win.
Talon He counters most AP mids, and Zilean is no exception. Consider a lane change, or just put your Greater seal of defense on and hope for the best
Ryze considering he would be building a Frozen Heart and a Rod of Ages he will be really hard to burst down.
Veigar No...
Vladimir your bombs stick onto his pool, and can do damage, but careful as you have no sustain and he is the king of sustain.(unless you count )
Ziggs dodge bombs. win.
Xerath No..
Zyra Don't run into plants. Win
Morgana Shield OP. Riot nerf
Zilean Your biggest enemy in lane is yourself
Brand dodge fire. win.
Pantheon No..

Zilean is annoying.