This is my AP Tristana Build. I haven't seen any other guides for an AP Tristana so I have made this one. Note that this guide does not advocate a pure caster AP build, it advocates more of a hybrid build, which is the build that I feel is most effective if you plan on playing Tristana like a caster.
The biggest advantage to playing Tristana AP is her burst damage. If you follow this build then you should be able to burst for 1.2k damage as a CONSERVATIVE estimate right when you finish Lich Bane, and still a significant amount of damage right when you hit Lv6. To my knowledge there is no other character who can burst for this much damage this early on, the closest competition being Sion.
Note that while you should be getting a lot of kills with this build, and you will be very dangerous as a ganker and after teamfights have been started, you CAN NOT initiate with this build and you WILL NOT be able to carry with this build. Your effectiveness will be at its maximum right around Lv16 when you get Lv3 Buster Shot and buy a Nashor's Tooth. It is just downhill afterwards as most everyone else will start finishing THEIR core builds and start stacking magic resist if they haven't already, and your damage will mostly stay static.
I do not advocate playing this build when your team does not have another hard carry. If you do, however, Tristana being able to burst people for virtually all of their health instantly is something that most people will not expect, and this build can be quite effective as a support caster.
Exhaust -- Ignite
Exhaust: Prevent people from getting away while you execute your burst combo, and possibly more importantly prevent them from being able to autoattack you in return
Ignite: Adds 200 almost guaranteed damage at Lv6, and 500 extra at Lv18. Usually overkill with the massive amounts of burst damage you will be doing, but with Presence of the Master making this only a 102s cooldown, there's no reason not to add it on to be sure of your kills.
- Sapphire Crystal
- Boots of Speed
- Lich Bane
- Sorceror's Shoes/Mercury's Treads
- Guinsoo's Rageblade
- Nashor's Tooth
- Zhonya's Ring
Those are the order that you should be buying and building your items in. You should start with Sapphire Crystal so you can be throwing out Explosive Shot more frequently and have a larger mana pool to Rocket Jump either on them or away from them as needed. Your next back should be Sheen and Boots of Speed. Back as soon as you have the money for Lich Bane, FINISH LICH BANE BEFORE SECOND TIER BOOTS. Rushing Lich Bane is the key to getting early kills and you will be able to afford better boots once Lich Bane GIVES you those early kills.
Whether you buy Sorceror's Shoes or Mercury's Treads should be a judgement call based on how much crowd control and how many hard disables/stuns the opposing team has. I usually buy Mercury's Treads instead of Sorceror's Shoes if at least 4/5 have crowd control OR if at least 3/5 have hard stuns/disables.
Lv1-5: Harass with Explosive Shot, once you have Sheen make sure you're combining Explosive Shot with an autoattack if you weren't already. If they have 50% or less health and are getting stupidly aggressive try to jump on them, Explosive Shot, Exhaust, Ignite, and then keep autoattacking them. Even at 5th level with only a Sheen you should be able to manage at LEAST 600+ damage with this combo.
Lv6+: You can now jump in for kills at even higher health, Buster Shot does 300 damage + (1.5AP) so being able to burst people from ~850 health down to kill them is a relatively safe assumption. Once you get Lich Bane (which you should be rushing) this number gets DRASTICALLY higher, assuming you land your entire combo you should be doing at least 1.2k in damage with the added AP and passive ability of Lich Bane.
This means that if you are in a side lane or with a teammate, you should be more or less getting guaranteed kills every time someone gets stunned around you and your cooldowns are up. In 1v1 or ganking, just be slightly more careful and try to only jump people a safe number under your damage burst, you need to go for sure kills because you are very fragile and if you screw up you will likely die.
Mid Game: After you have Lich Bane you should be roaming as much as the laning allows you to go get kills wherever you can, working on your Guinsoo's Rageblade. After you've got a Rageblade the game should be more focused on teamfights than laning if it isn't already. If people are still laning then you should be strong enough to jump anyone who can't hard disable you and kill them outright, so keep doing that until other people leave their lanes to deal with you.
In teamfights at this phase focus on keeping your Rageblade stacks up so your combo does as much damage as possible, and wait for the fighting to start. Then jump on the opposing carry and burst him for vastly more health than he likely will have, and use rocket jump to get back out as soon as he dies. Repeat on the 2nd most important member of their team as soon as your cooldowns are up if the opposing team isn't running already.
Late Game: Once you have completed Nashor's Tooth you will have even more ability power and have no trouble keeping your stacks up at almost all times. You will also attack fast enough that you will do no small amount of physical damage, although not as much as if you were a carry. Get as many kills as possible, trying to get a Zhonya's as fast as possible, although if they are stacking a lot (100+) of magic resist a Void Staff may be a better option. Your focus is still on helping your team by bursting down the dangerous members of the opposing team, the longer the game drags on you will only get worse at this, so it is important that you get enough kills early to where the enemy team cannot deal with you and your team, and the game does not reach the point where you start to decline.