Hi, I'm Gobmas, and this is my guide and build for AP Tristana. Yes, AP is an unusual thing to build on Tristana, but don't let the unusual aspect deter you. AP Tristana is a very viable and fun way to play in normal games. One thing I have to stress about this, though:

THIS BUILD IS NOT MEANT FOR RANKED!

I don't suggest playing this in solo queue ranked matches unless you are EXTREMELY confident in your team's and your own abilities. It is, in my experience, far too dependent on good teamwork to work in situations where your team is unreliable.

Pros:

Incredible harass game in lane
Very large amounts of single-target burst damage
Can easily chain kills in teamfights, often getting triple, quadra, and even pentakills
Nearly impossible to pin down
With some experience and an eye for opportunity, very easy to get fed

Cons:

Has a harder time last hitting
Squishiest thing since Anivia, so extremely vulnerable to CC
Long base cooldowns, meaning that blue buff is extremely important
Requires a different playstyle than most other champs, which can be hard to get used to
Not always a Con to some people in normal (hint: it usually is), but you are good at accidental Kill-Stealing

These are what make AP Tristana viable, like any other caster: her spells.

Draw a Bead: Largely useless outside of the laning phase. In lane, though, this ability is decently useful at getting in a little more damage with her auto attacks.

Rapid Fire: Helpful for proc-ing Lich Bane later on and for when you want to be dealing a bit more damage to a turret. But, like her passive, it is largely useless beyond that.

Rocket Jump: A very good AoE nuke that works almost like a massive flash, and it even cools down whenever you get a kill or assist! The slow is not often relevant, as you should be using this ability for the nuke, but don't forget that it is there. Also, while this can jump almost any wall in the game, it cannot jump the thicker walls in Summoner's Rift, like the ones near mid.

Explosive Shot: This active is Tristana's main source of damage in her lane, and it is an extremely efficient source of harassment. The max rank damage on this skill may not look like much, but the early game damage is actually very high, so with some early AP to complement its 100% AP scaling, this can dish out some hefty damage. Try to get in as many hits with this in the first 4 levels as you can, as that is when it is at it's most efficient. The largest downside to this ability is the short casting range, shorter than her attack range at all but level 1. This means that some movespeed is mandatory to be able to dominate your lane with this spell. Oh, and the passive is nice for farming, although you should be careful about how much it can push your lane.

Buster Shot: This ability is why AP Tristana is so viable. The 150% scaling and large base damage make for one of the largest nukes in the game, and certainly the most reliable. With a Rabadon's or a fully stacked Mejai's, this single spell can almost 1-shot squishy champs, and the low cooldown means that you can do that A LOT. The knockback is only icing on the cake, providing just one more way for you to get out of a teamfight or gank without a scratch, or preventing a counterattack as the DoTs from Explosive Shot and Ignite finish your victim off.

The runes are largely self-explanatory for casters, but everything can be changed around for anything else that suits you. I use:
Magic Penetration Marks]], for a good damage increase at all points in the game.
Flat Health Seals, to help you take burst damage in lane.
Flat AP Glyphs, for a large damage increase in the early game. I highly suggest keeping these no matter your preferences, as the damage increase these give your Explosive Shot is invaluable in the early game.
Flat AP Quintessences, again for the large damage increase early game.

Alternatively, I've found that Movement Speed Quintessences can help increase Tristana's terrible movespeed.

What I Use:

Ignite: A great choice to guarantee kills, it's best in the early game. Not as good late-game for escaping enemies, because of its short range. This can be replaced easily with a spell more to your liking.

Flash: The best summoner spell in the game, by far. In conjunction with Rocket Jump, you are almost impossible to catch as long as this is up. Also good for positioning yourself to take advantage of the knockback from Buster Shot and to counter enemy Flashes.

Other Viable Options:

Teleport: Not as useful for backdooring on an AP champion, but it still has many other uses, like getting back to your lane faster or making sure you don't miss a teamfight.

Exhaust: Useful if you want to stop a fed Tryndamere from staying fed. Always a good summoner spell, but not as useful on Tristana as on less bursty champs.

Cleanse: As I've said, one of AP Tristana's main weaknesses is to CC, and this helps aliviate that problem. Best used in conjunction with Flash for maximum survivability.

Ghost: If you dislike the length of Flash's cooldown, this is a good alternative. Can also be used similarly to Teleport.

All the other spells are ones you shouldn't really be getting on AP Tristana. They all have their uses, but are probably better on someone else on your team.

The item build with AP is largely static, not much about it is going to change from match to match. The order you buy the items and their components should change depending on your priorities each game, however.

Boots: Although some people like to begin a match with a Doran's Ring, it is usually better on most solo lanes to get boots to survive against people like Annie, Orianna, and Brand. The large amount of HP you get from the HP pots also helps mitigate a large amount of harassment. Starting with a Doran's Ring is still best if you decide to go with Movement Speed Quints, though.

Mejai's Soulstealer: Only get this item if you are 90% sure that you will not die at all. It is fairly easy to get stacks and get out in teamfights, and the ease with which you can gank lanes also facilitates getting stacks early on. BUT BE WARNED! Dying even once can mean an end to your damage for the rest of the game, since you will have a hard time catching back up.

Rabadon's Deathcap: This is what you should be rushing most games, especially if you haven't tried AP Tristana before. The amount of AP you can get from this single item is just staggering, and once you have it, you should be able to instantly kill those without defenses.

Lich Bane: This item will add huge amounts of damage to your already large burst, as well as helping circumvent the early MR that your opponents will inevitably pick up. The movespeed, small amount of MR and max mana all contribute to an overall great item.

Deathfire Grasp: The burst and cooldown reduction, plus a small amount of AP, will make kills as easy as w-r-1. Really, even offtanks should be scared of you at this point. If the enemies are stacking MR, however, it might just be better to go straight for your Void Staff.

Void Staff: The 40% Magic pen and AP will only add to your huge damage, although this is generally best if the enemy carries have caught on and started buying lots of MR.

Guardian Angel: If you are being focused by a lot of CC and can't even kill someone without getting jumped on in the process, picking up one of these is greatly helpful. This way, even if you get focused after jumping in and getting a kill, you can still easily jump right back out again. Also has good synergy with Mejai's Soulstealer.

Farming

Farming can often be difficult with Tristana, especially when it is so easy to harass constantly and just forget to last hit some minions in the process. But if you have to choose between one or the other, ALWAYS GET YOUR FARM. Even if you are bad at last hitting, keep trying, as practice is the only way to get better at it. Once you get later in the game it gets easier, as rocket jump should one-shot all the minions in a wave and the passive of explosive shot will make picking up the rest easy. But whatever you do, don't neglect your early farm.

Early Game

Go mid if at all possible, although solo top is okay as well if you both have junglers. If you have to lane against 2 people, it is far better to have a partner, though, so don't shy away from bottom if you have no alternative. If you did get a 1v1 lane, though, congratulations! This is where AD Tristana often has trouble, with low damage auto attacks and no great harassing skills. YOU, on the other hand, are a great solo laner! With decent luck/skill at dodging skill shots, you will easily get a kill in lane by at least level 6. Just harass with Explosive Shot whenever it is off cooldown, then run back, kiting with a few auto-attacks if they give chase. When they are low enough, Rocket Jump on top of them, and blow your cooldowns. All of them. Gratz on your kill!
Now get back before minion aggro or the enemy jungler get around to picking you off, and go get your finished Sorcerer's Shoes along with the pieces of your first major item and some wards/potions. You don't have to go straight back to mid, as Tristana is a great ganker; feel free to go gank some sidelanes if the opportunity presents itself. But always make sure that you are keeping up in minion kills, as well.

Mid Game

By now some small teamfights should be starting up; take this opportunity to sneak in some kills and/or Mejai's stacks. And by sneak, i mean SNEAK. Your role in teamfights is twofold: stay out of them, at least at first. Wait for their carry to expose themselves, and maybe even take some damage, then Rocket Jump on top of them and, once again, blow your cooldowns. Even if you have to turret dive to get at them, do it. Once you get the kill, your Rocket Jump will refresh, letting you jump back out again. Be careful in this process for members of the enemy team such as Akali, Kassadin and so on attempting to do this to YOUR team. If you run into them, just run. You don't have the sustained damage to actually kill them at full HP yet, but you should be able to get away without dying unless they are already fed beyond imagining, and maybe even kite them into your team.
Once the carry on their team is dead, you should have no trouble spamming Explosive Shot and chaining Rocket Jumps onto multiple dying enemies. Just be careful not to killsteal from other carries, as Tristana is very good at accidentally picking up kills when actual carries did all the work. If all goes well, you should be able to finish your Rabadon's Deathcap and Lich Bane in short order and move on to late game.

Late Game

Now the game should be all about teamfights and pushing towers. Follow the same basic tactics as in the mid game, and finish your Deathfire Grasp as soon as possible. It should also be pointed out that with a Clarivoyance or ward, Tristana can easily steal Baron or Dragon by Rocket Jumping in and Buster Shot-ing the epic monster. This should also knock back the enemy team in the process. Decide what your last items should be, don't 1v5, and have a GG.