Now, before you freak out about the 3 Doran's Rings take a moment and consider what this actually does. Off the bat you have 5 bars of health, enough Mana Regen (with mastery of +3 m/5) to allow you to spam just about as many Jack in the boxes as you see fit, and a nice +15 AP boost.
And with only 950 more gold (fairly quick in 3v3) you can triple that with two more Doran's Rings. The boost of mp5 will account for the increased cost of Jack in the boxes, and allow you to spam your Shiv. The Health and AP are nice to have, you will not only be hard to kill, but also be able to spam your hard-hitting abilities.

I add many "( )" Parenthesis in my sentences, and then proceed to finish the sentence afterward. This can be confusing, so please read carefully. All the Parenthesis are important, and often contain key tips.

Not many champion builds call for ALL AP RUNES. However, since this is 3v3, and the matches tend to be rather short, it is important to start strong. With near 50 AP at the start you will be a force to be reckoned with (Read the "Early Game" section to start off the right way).

Marks: Greater Mark of Potency (AP), or Greater Mark of Insight (Magic Pen)
Seals: Greater Seal of Potency (AP)
Glyphs: Greater Glyph of Potency (AP)
Quintessences: Greater Quintessence of Potency (AP)

If you feel another rune would be better, feel free to explore.

Fairly Standard Masteries, Good M/p5 Boost, Magic Pen, Cooldown Reduction. Not much else to say here. However, if you pick other summoners spells adjust your masteries accordingly.

I would first like to point out that the order of items above is in no way the order you need to build. In fact, it's really just a list of usable items (after Deathcap) than an order.

Here is where things get tricky. I almost always get 3 Doran's Rings and then a Rabadon's Deathcap after that. That's your core AP damage. After that depends entirely on the game. (FYI I recommend not selling any of your Doran's Rings until you at least have something to compensate for the health loss. (Rod of Ages, Rylai's)

If you are destroying the enemy team you could get a:

Rylai's Scepter - I don't know if it effects boxes, but i believe it adds to Shiv slow (unconfirmed). Either way it adds some good health and AP. (If it doesn't add slow then Rod of Ages would probably be a better replacement)

Majai's Soulstealer - Never actually used this myself, but I can see how it would be useful, I would suggest getting deathcap first.

Any of the other listed items would be good at this point. If you think they may start to overpower you, perhaps try a Frozen Heart or other enemy-appropriate item ( If their Annie is getting fed, Frozen Heart would not help you)

If you are about even Kill:Death ratio try a:

Rod of Ages - The bonus Health will help your survivability, the AP will help you hit more, and the Mana will make sure you never go oom. Not to mention the increasing effects over time and Health/Mana boost every time you level.

Rylai's Scepter - Again, this item is going to give you that health you need, and may help you escape/get the kill with bonus Shiv slow.

If you are getting ganked/getting pwnd try :

Boots! - Boots, Magical Things. I usually don't get them, unless the match goes past 40min. However, if you are getting owned, they can really help you get around. Many different types of boots will do: Swiftness, Magic Pen, Cooldown, Merc Treads.

Rod of Ages + Rylai's Scepter - Both Rylai's and Rod of Ages are great for 3v3. As the game goes on everyone should be getting stronger and harder to kill. That applies to you as well.

You may have noticed Rod of Ages and Rylai's Scepter both showed up in just about every case. Both have tremendous stats and can set you above other Shaco's Building Mejai's off the bat (This leaves you squishy and probably wont work for high elo matches).

Starting with Jack in the Box is great for setting up traps in the bot area/top lane. Place them in any of the three bottom-middle bushes, or where the Lizard will eventually spawn (Red Buff). You may want to stack multiple in a single place, or spread them out. People will likely beware of entering these areas when they see the other team has a Shaco (Although they may assume your AD Shaco and be less wary). Placing them in unusual spots may fool players into thinking it is safe, but it's less likely they will go that way (People often go the quickest way though jungle areas).

Another way to start is to get Two-Shiv Poison. It should hit around 100 damage at Level 1 (not including MR/Armor) and provides a nice slow. This can help keep someone from chasing you when you rush bot.

Level 4 - Getting Deceive is gonna let you Teleport through jungle, sneak up on people, and hit a bit harder w/ a slow from Shiv.

Level 6 - Ult Time! After popping this you will be able to auto-attack twice as much, have a clone that explodes for some serious damage, and fake stunt clone to distract the enemy and make them blow their cooldowns.

Ignite - Nice damage, good early game kills. Can stop lifesteal/health regen champs from healing if they dive you and your boxes.

Flash - A second Q!. If your Deceive is on cooldown, you can use this in place, or combine them for a huge jump. great for getting ahead of people your chasing, or getting away through jungle.

Other Spells:

Exhaust - Not my choice for AP, but certainly can help stop someone from getting away, or to hurt someone's damage output. Certainly Viable.

Ghost - Not really that good in my opinion, doesn't help much if your slowed or stunned. If you feel like you need ghost you could try it.

Typically I will grab my Doran's Ring, rush bot, and place my boxes in the bottom 3 bushes. 1st one in the bottom bush. If they don't show, ill put them in the 2 bushes directly above it. Help you teammates with your ignite and boxes, and even some auto-attacks if you can. Even though you are AP Shaco you still have AD champ-like base attack damage (115 base attack damage at 18). Use your flash to get out of a death, or to catch that guy who is about to die (Beware, you have little damage at Level 1 besides boxes, which take around 2-3 seconds to activate!) However, a full box going off on 1 champion can easily bring him to half health or below, so place wisely! The aoe fear from the boxes (not very big range though) can help take the pressure off you and your team, while putting the hurt on your enemies!

After the initial rush you can take either:

Top: A rather risky move, if your enemy can harass you or catch you without your boxes. Putting boxes in one of the two bushes (Make sure you put it as close to the bottom wall as you can, and stack them together as close as you can! The bushes are small, and they have a good chance of targeting minions as they walk by) preferably the one on your team's half of the map (Bush closer to your Nexus). Try to lure your enemy champion (At Level 2 you can harass with shiv, usually forcing them to attack or flee) into you stack of boxes. If the Champion is melee you can run up and auto attack him or the enemy minions (Foes like Jax will "jump" at the opportunity to hit you). If the Champion is ranged, it may be best to wait till Level 2 were you can counter range harass (Shiv)

Bottom: You may end up taking this lane more often then top, as your AD champ may well prefer top. Simply place boxes in the top two bushes to stop enemy ganks back and forth, harass with Shiv. Depending on who you are with, you may need to protect them, or help them get the kill.

I consider Mid-Game in 3v3 to be around the time your get your Needlessly Large Rod and Blasting Wand (right before you finish your Deathcap). By this time you hopefully have been able to use Deceive to gank the other lane (This works great if you are bot. You can jump from Lizard area to Top Lane Bushes, place a Box and throw your Shiv/Ignite, easy early kills.), and get some kills. If you were bogged down in your lane, or haven't ganked for some other reason, then now is the time. You can place boxes at your turret to protect it from pushes, and then go on your way. Once you get Deathcap you will have a NICE chunk of AP (I think its about 200 but I can't recall exactly). Your Shiv gets 100% of your Ability Power! And even half your AD. This is why Hextech Gunblade is a recommended item, it can help hit hard with Deceive and Shiv. The Dmg-Slow is just another plus.

If the game hasn't ended yet then it's about time it should. You should be hardy by now, with plenty of health and AP, and maybe even some defense. Shaco can backdoor like a pro, that is, if there aren't any towers. That might seem like its not that impressive if there aren't any towers. Well, think again. He can stealth up to the Inhibitor, Drop a Box + Pop his ult. In mere seconds after his enemies can even see hes there, the turret is down, and Shaco is free to go where he pleases. Should they find you before you down it, you have a clone decoy to finish the tower while you run, or to distract the enemy while you finish it. Fleeing is simple once the Inhib is down, Flash, Decieve, place Boxes behind you. You should be just fine.

- Jack in the Boxes can attack Turrets and Inhibitors! However, if there are minions or Champions in range it may attack them instead.
- Most Abilites wont be able to hurt your Boxes (Such as Malzahar's "Malefic Visions" his Single-Target DoT) From my experience it seems that abilities that do Attack damage, not Magic Damage, can hurt boxes (I believe Tryndamere's Spin and Ezreal's Ult both hurt boxes, but so does Malphite's Ult, which Is Magic Damage. The difference seems yet unclear.)
I have noticed if you are standing on a box, and you get hit by an aoe ability it can hurt the Box. It seems to only happen if you are directly on the box (A Glitch I believe)

As I am playing, I will likely think of more tips. I will add more over time, so check back in every once in a while to see.

All in all my guide is intended to be exactly what the title says. AP Shaco, with good survivability from start to finish. If you want to just own people on a smurf account, and build Mejai's off the bat so you can get fed stacks, this probably isn't for you. You certainly CAN get fed with my guide, but you wont die every time you try to gank someone, and another champion comes to help him. With you Ult your survivability sky rockets, and so does your damage. I can't tell you how many times my clone has got a kill because the enemy team killed him instead of me, and I walk away Scot-free. I hope you liked my guide, please feel free to comment below with anything you have to say about the guide. Happy Killing!