Zilean is a great support champ and he can deal quite a lot damage too if built properly.
Zilean's job at the Fields of Justice is to look after his teammates: give them Time Warps (speed boost) and cast an Chronoshift (resurrect) on dying teammates.

I've been playing Zilean for quite a long time now and I've tried several different builds, but this works the best for me. Don't expect that if you follow this guide you will rip the enemy team in half. Zilean is a support champion and he can't win the game alone. Zilean has one damaging ability, Time Bomb. Zilean can't win on his own, but if you have a good carry by your side, you can make the carry even more dangerous and help him with time warps and ultimates.

Now for the build+game strategies, don't judge me, this is my first self-written guide and my English is not perfect. Hope you all still understand.

Pros:
Good speed boost on friendly champion as well as good slow for enemy champ.
The two-bomb combo can destroy enemy champions quickly.
Good late game farmer with time bombs.
Cooldown reduction skill that helps a lot.
Ultimate helps a lot in team fights.
The items in this build give you plenty of hp so you can survive better and stay alive to support your team.
Experience bonus for your team!

Cons:
Mana hungry
Slow early-game farmer
Hard to master (requires training)
Vulnerable to CC

Early:
Zilean can be played as mid champion but Zilean fits better into lane fighting than mid because Zilean don't benefit from huge farm as much as DPS champions. Let a DPS champ go mid while you support your lane partner and your mid gets better farm. Of course if your team has no mid hero or someone doesn't want to go mid (example if he has a premade lane partner) you can always take the challenge and go mid. Doran's ring))Doran's ring is a good item to buy first because it gives you some hp, mana reg and 15 ap comes handy while harassing with bombs. At start Zilean should be played passively, just last hit minions and harass enemies if they try to kill the minions. At lvl 6 you can try to score a kill. No matter if you fail at this, because you have your ulti ready. Just make sure you don't run out of mana if this happens. If you picked [[Flash]] as your summoner spell, this works great: Flash+ Time Bomb+ Rewind+ Time Bomb+ Time Warp on the enemy+ Time Bomb always surprises the enemy champion and often can provide you a kill or assist if your lane mate was ready too. This can be done also without Flash like this: Time Warp on yourself+ Time Bomb+ Rewind+ Time Bomb+ Time Warp on the enemy+ Time BombThis doesn't work if the enemy champ has flash too unless your lane mate has a stun to counter it. You should now buy ((Ionian Boots of Lucidity]] because Zilean is very dependent on cool downs and cdr will help the gaming alot. AND don't forget that support champions must buy Sight Wards and Vision Wards when necessary so warding is your responsibility when you are playing Zilean. Good spots are of course: Dragon, blue buffs, red buffs, river bushes.

Mid:
As first "proper" item could be either Mejai's Soulstealer or Soul Shroud, depending on the situation (explained later). Depending on your summoner spells, if you chose Teleport, farm minions and defend towers. At team fights, stay back and let tank and dps-champions go first. Keep an eye on your team's best carry and be ready to cast an Chronoshift on the champion.

There is nothing more important you to do than resurrect the carry.

So don't drain your mana low by spamming bombs and rewind. Time Warp on the enemy carry can make the fight much easier for your team. Cast time warp on enemy champions that are trying to flee the battle on low health so your team gets more money from kills+assists. Blue buff is useful on Zilean, take it always unless your team has a fed Kassadin or some other mana hungry dps.

Late:
Now if your team has done good, you should have at least 3-4 of the items mentioned earlier on the item section. The late game goes same as mid. Focus on the best carry on your team and keep him alive with your ulti. In the late game, example when doing baron. You can't deal damage to baron very well so you'd better scout the nearby bushes in case if the enemy team tries to assault there when your team fights at baron.

Ultimate using:
Always when your ultimate is on cooldown, spam rewind so your ulti will recharge faster. Ultimate should be cast on a dying heavy carry on your team so when he comes back alive he can continue slashing the enemy team.

Teleport - Good to have, makes your movement faster and helps a lot to defend towers.

Flash - Escape mechanism as well as chasing mechanism. I mentioned before the strategy to get an easy kill on the lane. Flash+Bomb+Rewind+Bomb

Clarity - Good when your team has many mana hungry champions. Picking clarity is a good way to support even more.

Clairvoyance - Good for scouting and provides more supportivity.

Fortify - Useful in some games, when enemy team tries to backdoor towers. Counters Shacos backdoors with his ulti etc. Sometimes this might be very handy, sometimes not.

I usually go with Teleport+ Flash or Clarity+ Flash.

First item: Doran's ring (hp, mana regen, and ap)

Boots:
Ionian boots are imo the best because the cdr helps a lot.
Boots of swiftness are another option for the movement bonus. You can go wichever you like more.
If enemy team has lots of cc, go for the Mercury treadsMercury treads.

Core:
Soul shroud (lots of hp and an awesome aura: cdr and mana reg)
(hp and mana and boosts your ap)
Rylai's crystal scepter (lots of ap, health, and most important: the slowing debuff on enemies)
These three items should be bought always.

Optional:
Mejai should be bought before soul shroud in those cases when you don't get killed many times and you get lots of stacks to it.
if enemy team is heavily armed with magical damage.
Zhonya's hourglass if enemy team has lots of physical damage. The "Stasis" ability in Zhonya can save you sometimes if it is used correctly. Replace Abyssal Scepter with this when necessary.
Rabadon's Deathcap if you need more damage and not so much suvivability. Boosts your time bomb's damage high up the sky.
Because Zilean is vulnerable to CC, in some cases you should buy Quicksilver sash. Example when enemy team always stun you when team fight starts and you are unable to look after your team's carries when stunned or silenced. It doesn't cost very much so you can buy it pretty much in any time you want, if you are a good player you know when is time for this.

Cheat sheet:
Ionian Boots of Lucidity, Mejai's Soulstealer, Soul Shroud, Rod of Ages, Rylai's Crystal Scepter, Abyssal Mask/ Zhonya's Hourglass
And if the game lasts this long you could sell the boots and buy Rabadon's Deathcap if necessary.

Zilean has few nice tricks that can come handy in different situations.

The 2-bomb trick:
Time Bomb+ Rewind+ Time Bomb This combo can be done by three bombs after you get some CDR so you can cast rewind faster after the 2nd bomb you planted, this way you can bomb the enemy three times in a short period.

Invis trick:
If you are laning with Teemo, Evelynn, Twitch or Shaco: Time Bombs can be planted on an invisible frienly unit, it's a good way to harass enemy team. Cast Time Bomb on your lane partner and he runs next to enemy champions while being invisible. Also works with non-invisible champions such as Jax, Tryndamere, Shen who can jump towards enemy champions.

Ward trick:
Buy a ward and plant it in the area where is action (depending on which lane pushes more or if the lanes or if both lanes stay in the middle area of the towers. When the ward is planted, you can Time Bomb the ward and enemies near the ward will get surprised by the invisible damage.

Zilean vs. Zilean trick:
I think not many of inexperienced Zilean-players know this one. If you play versus another Zilean and he nukes you with his bombs, you can cast Time Bomb on yourself and it will remove the enemy bomb. With this trick you can prevent enemy Zilean from getting kills while you are low on health or just prevent them of harassing you. You can also defuse enemy Zilean's Time Bombs from your team mate by using this trick.

Rewind spamming trick:
When Chronoshift is on cooldown, keep spamming Rewind so your ultimate recharges faster. I often hear frustrated enemy players rage: "OMGGMG how you have ulti all the time cheater! Go **** your ******!!!!??" This is important and trust me, spending mana on rewind continuously is worth it when you have ultimate ready almost in every team fight.