READ THIS GUIDE IF YOU WANT IT TO BE EFFECTIVE!
The cool thing about Mobafire is that you can see not only the runes, masteries, stats at level 18, and skill orders, but also the guide writer's analysis. I wrote a good deal of information and opinion about this champion. If you're new to Poppy, or if you want more insight into the champion, please read the rest of the guide. It took a decent length of time to write, and I thought about every word I put down.

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What happened to AP Poppy? I'm going to share with you my past experiences with Poppy; you can probably skip this section if you want. When I first picked up the game, I had no idea who to play. I was trying all of the free champions, and I hated them all. I wasn't getting into the game successfully. I asked my friend, a level 30 who introduced me to League of Legends, who I should play, and he told me Poppy. Back then I didn't know it, but he later told me he recommended Poppy because I died a lot and Poppy isn't a champion who dies easily. I had no idea how to build anybody, so I went AD on her. I probably did pitiful damage, too. I was sub level 15, and I had no idea how to build or last hit or gank or do anything useful. Anyway, my team and my enemies always made fun of me because I went AD on Poppy. I guess the trend was just AP. After taking a long hiatus from the game, I switched to other champions. When I finally came back to Poppy, I asked a different friend what I should build, and he recommended an AP build to me. Over tons of games, I have decided on a build that works great for me, and I think it could be great for others too, if they tried it. But the thing is, I never see AP Poppy anymore. I keep seeing the same build that starts with 2 Philosopher's Stones that allow enough farm for Trinity Force. Yes, Trinity Force is viable, yes, it does great damage, and yes, AD Poppy is a great build, but it just seems odd to me that everybody dropped the build that I have the most success with.

You can look at the tree for masteries, but the important ones to hit are Archaic Knowledge (the spell pen), Awareness (experience), Quickness, and improved Ghost and Teleport.

For runes, you can probably go with whatever your normal caster runes are, but the Movement Speed Quints are VERY IMPORTANT. The thing that's great about Poppy is how fast she can go. Her W gives her a speed buff of up to 25%, Lich Bane gives her 7%, and Ghost gives her 27%. An additional 4.5% MS compounds on that making her inescapable, especially if she has her ultimate on somebody without any hard CC.

I take Ghost and Teleport on Poppy. For the reason above, Ghost is mandatory, but Teleport is just a personal preference. I use it because Poppy has absolutely no AoE damage, so I like to teleport to minion mobs to get some precious farm. It's also good for defending and even backdooring. Poppy can backdoor very well even while going AP because Lich Bane will proc on the tower when you use your W, which gives you AD. In addition, and this works with any champion who has a spell that resets his/her autoattack (Nasus, Blitzcrank), you can use your Q immediately after attacking the tower and you will instantly hit the tower again. It doesn't proc the bonus damage of your Q, but it will once again proc Lich Bane. I like to hit W, smack the tower a few times, and hit Q immediately after autoattacking about 2 seconds later to get another Lich Bane proc. At the end of the game, if you farmed well and got good stacks on Mejai's, Lich Bane can deal around 400 damage to a building! That's about a quarter of an inhibitor's hp!

Other viable spells include Flash and Ignite. I wouldn't recommend anything else. Flash is obviously an amazing escape mechanism, but it can also help you the same way it helps Alistar; flash to behind somebody you want and push him into your tower/teammates with your E. Ignite can be useful the same way it always is. Make sure that guy dies, cut his healing in half, whatever. I just personally find the most use with teleport.

EDIT: I've actually started taking ignite over teleport. Tele can always be a handy skill, especially on somebody with high move speed on Poppy because it lets you defend a tower and then chase whoever was trying to knock it down, but I just recently realized what makes ignite amazing on the Iron Ambassador: it works with your ultimate! Ignite deals true damage, and her ult increases all damage Poppy deals by a percentage, so if you ignite the same target as your ultimate, you're actually dealing a nice chunk of TRUE DAMAGE to your opponent.

Here's where it gets good.

Passive: Valiant Fighter: This is arguably one of the best passives in the game. It's not something that you will notice all the time, but it's working behind the scenes to keep you safe. Any time you take damage that exceeds 10% of your current HP, that damage is decreased by 50%! If you think about it, that can save you a lot! This doesn't mean tower dive at level 1 (the passive does NOT decrease damage by buildings), this doesn't mean play absurdly aggressively, and this does not mean tank the team 1v5. It just means you're deceivingly durable.

Devastating Blow: Makes your next basic attack deal 20/40/60/80/100 base damage(+0.6xAP) + 8% of the target's max HP. The bonus damage can't exceed a certain amount, but I don't even really know what this means. Costs 55 mana at all ranks.

This move is half of your burst, and it's a great tool for either farming or killing. It costs a constant 55 mana, so feel free to use it often. It also has an amazingly small CD. If you have blue buff and Deathfire Grasp, you can smash with it about every 2 seconds. The best part about it is that since you need to autoattack to use it, it procs Lich Bane. When your AP starts picking up, it does massive damage.

Paragon of Demacia: Gives 1.5/2/2.5/3/3.5 AD and armor every time you hit an enemy with a basic attack or get hit by an enemy with a basic attack. Stacks up to 10 times. Using the active gives you 10 stacks for 5 seconds while also giving 17/19/21/23/25% movement speed. Costs 70 mana at all ranks.

This spell is great for several reasons. It lets you initiate, escape, or just be fast; it gives you a decent AD/armor buff that lets you last hit a little better and makes you a bit tougher; it sets up your Lich Bane so you can have another heavily empowered attack. Watch out though because the 70 mana needed to use this can drain your small mana pool early game.

Heroic Charge: Click a target to tackle him and push him, dealing 50/75/100/125/150(+0.4xAP) damage. If you collide with terrain (anything from walls to trees to even sometimes minions), the target gets stunned for 1.5 seconds and an additional 75/125/175/225/275(+0.4xAP) damage is dealt. Costs 60/65/70/75/80 mana.

YES. This is what makes Poppy golden. Not only do you get all the utility of pushing the target where you want him (think Gragas ultimate or Alistar headbutt), but you also get to deal ridiculous damage and STUN YOUR TARGET. This spell causes Poppy to dash to an enemy champion and push him a small distance. If the two of them collide with a piece of terrain, the target gets stunned for a second and a half, letting Poppy destroy him with her Q and letting the rest of the team beat down on him too, if Poppy can't handle him alone. The move has decent base damage and decent scaling before the collision, but more base damage is added if and when you do collide with terrain, and the AP scaling is added back on too!

Diplomatic Immunity: Poppy marks a target enemy champion for 6/7/8 seconds. While this is up, only that champion can damage or CC Poppy. Poppy also deals 20/30/40% increased damage to this target during the duration.

YES. This is also what makes Poppy golden! Be prepared to increase your burst by 20/30/40%! And while doing that, make yourself immune to 80% of the team! When you pop your ultimate, you can only be touched by the person you throw it on. You can't take damage or be crowd controlled by anybody else! This allows for infinite strategies. I have a lot to say about this move, so I'll do it in the strategies section.

Start with a Sapphire Crystal and 2 red pots. Poppy has a tiny mana pool, so you need the Crystal not only to help with that, but also to get you closer to Sheen. Your Q does base damage plus your autoattack plus a percentage of the target's max HP plus the AP scaling, so increasing any of these increases your damage. Sheen is the most cost-effective way to multiply your damage output early game. I usually go bottom lane (you're going to get harassed a LOT if you try to solo top or mid) and try to get enough last hits to reach 860 gold. At this point, unless my lane partner needs help badly, I go back, buy Sheen, and teleport back to my lane. It's like I was never gone, but now my Q does stupid damage. At this point your Q can nearly one-shot caster minions, allowing for much easier farm.

Your next item is Sorc shoes because they're great on any caster. I suppose if you're going against a lot of CC, you can get Merc Treads, but Poppy can nullify 80% of any team's CC, so it might not be necessary. I usually just pick Sorc shoes no matter what because I'm greedy for damage.

Your next item is Kage's Lucky Pick. It helps a lot because like I said, Poppy has no AoE skills to help her farm. Gold per 5 is a great way for her to earn money even though she gets harassed easily early game and can't damage more than one enemy at a time. Plus it builds into DFG, and the CDR and damage from this item is excellent. When Morello's Evil Tome first came out, I thought about switching, but I can't live without the period of the game where I have extra Gold per 5.

This is where you have options. Are you stomping them? Are you landing every tackle, hitting every Q, and scoring kills left and right? Get your Mejai's Soulstealer now. Are you having trouble staying alive or dealing damage? Finish off Lich Bane. You can get Mejai's as early as after your boots or never at all. I like to get it even if I'm not doing so great because I tend to rack up assists, and Poppy almost always gets away with her ultimate/Ghost/W.

After you have boots, Lich Bane, Mejai's, and Deathfire Grasp, it's time to massively increase your damage: get Rabadon's Deathcap. You're really hitting hard now. You should be able to kill any squishy in 1 burst. Some it might only take 2 of your 3 damage dealing spells (I'm including the DFG active) For those of you who don't know, Deathfire Grasp deals 30% of a target's current HP as magic damage (that means use it first), but it also scales on AP; for every 100 Ability Power you have, DFG's damage increases by 3.5%. On top of that, your ultimate increases ALL damaage by 40% at maximum level! So if you pop your ultimate before throwing Deathfire Grasp, you can do somewhere around 70% of somebody's HP with an item! Long story short, use DFG's active. It's too good to pass up.

I can never decide what to get after Rabadon's Deathcap. It's rare that you can earn enough money to get anything expensive after it. I've tried a lot of funny stuff: Rylai's so that my Q slows, Hextech Gunblade so that I get roughly 1/3-1/2 of my health back by bursting, Guardian Angel to be a massive troll. The 6th item is completely situational, but don't expect to finish your build very often. Usually if I'm doing well enough to get the Rabadon's, I'm something like 10/0, and teams seem to like to surrender when somebody is legendary and hasn't died yet.

Tl;dr
Sheen first, boots, Lucky Pick, Mejai's, Lich Bane, DFG, Rabadon's, ???

EDIT: I recently started trying the two philostones build starting with a Regrowth Pendant and one potion. I would go back at about 1000 gold to finish the first stone and purchase a second, and then I would continue farming bot to get tier one boots and Sheen. The rest of my build was the same, but this build works for a few reasons: having three gold per 10 items instead of one is INSANE, and the regen you get from the philostones is also ridiculously sweet. I would also build one philostone into the new tenacity item to keep into late game, but I would sell the other for AP after it has made its use with the gold. Try this if you want a more stable early game.

Early game: Focus on last hits. Do your best to get 860 gold ASAP so that you can go buy your Sheen and get back to lane. Sheen will make it way easier to last hit, and you will deal insane damage to anybody you can get your stun on. Try to lane with somebody who can zone your opponent so that they are pushed to a wall. Then you can tackle and secure a kill.

Mid/late game: I like to run around the map like a crazy person trying to gank and get kills. If it's ok with your jungler, blue is pretty good on Poppy. It lets you use your Q a lot more often, and it lets your ult have a low CD so that you can go into teamfights more often. You can play however you want to, but I like to locate a squishy and play as an anticarry.

Using your ultimate: There are tons of way to effectively use Poppy's ult. I like to ult the squishy so that I do increased damage. If you can take out somebody like Kassadin who can do a lot of damage, the rest of the team might not be powerful enough to kill you, so you won't need your ult for the rest of the battle. AD Poppy needs to be more conservative with her ultimates since she doesn't have the burst, so she needs to be in the fight longer. A good strategy is always to ult the person who deals the least damage or has the least CC. Just make sure you don't R somebody and then you aren't able to catch up. It's terrible to waste her ult when it can be so useful.

In any case, here's how your burst should go. Ult if you have it (either the person you want dead or the person who won't hurt you much), then DFG, then tackle, then Q. The good thing about tackle is that it always allows you to land an auto with Q. The bad thing is that sometimes the game mechanics make you go hit a minion or something, which is really stupid. I guess you can right click the target to ensure you hit your mark. Just try to always hit your stun because it greatly improves your damage.

I hope everybody who read this learned something. I think more people should play AP Poppy because she has one of the highest burst potentials in the game. I can't think of anybody else who can easily kill somebody from full health with two spells. I am by no means an authority on the game, or even a high ranking player, but I do well with this build. Try it out if you want to play somebody new or if you want to try a new build. Thanks for reading!