A little bit about myself; I enjoy unorthodox character roles that work. I find it amusing to build attack speed and take full advantage of Lulu's passive, head down mid and abuse Sona's lane bulying capabilities, etc. and as such am presenting to you AP Jinx.

AP Jinx is not AD Sona (A character doing a role just because they can, no matter how bad they are at it) because of four things:

1. Her Flame Chompers scale off of AP. In fact, they have the same AP scaling as Annie's Tibbers. (100% AP) They have less base damage and don't summon a Tibbers, (Tibber's initial burn damage is included in that comparison mind you) but they reach ~10 second cooldown and can damage an enemy champion with each individual chomper, depending. They also always stun.

2. Her Switcheroo ability gives attack speed or range and splash damage. The former allows her to reach amazing DPS (Higher damage output than AD Jinx earlier on, if I'm not mistaken) and the latter allows her to lane harder. These can be utilized and even capitalized on with an AP build.

3. Not only is no one expecting an AP Jinx, but you even disguise yourself as AD early-game. This can really throw off those building armor.

4. You zone like a boss. In bot lane you have to deal with two people. In mid lane, you can be far more manipulative.

With that being said I wouldn't call her an entirely legitimate choice. Still yet, I find it fun and more stable than AD Jinx, even if not as powerful late-game.

As a side note, AP Jinx is actually an APAAC (AP auto-attack carry - like Teemo) and slightly mixed rather than pure AP. (She just has so much innate AD and you benefit greatly from the tacked-on AD Guinsoo's Rageblade offers) Half-way through the game, you're likely to have the same amount of AD as AP.

I choose to start out with a Doran's Blade for the sustain and the fact that it's technically an on-hit item. You thrive off of attack speed, so you benefit from this more than an AD Jinx.

Health Potions are helpful too.

Sorcerer's Shoes are pretty much what they say they are. Take these over Berserker's Greaves because the build already offers all the attack speed you could ever possibly need.

Nashor's Tooth is pretty much essential to any APAAC. It provides both attack speed, AP, and on-hit damage that scales off of AP. To top it all off, it gives you some cooldown reduction with which you can zone harder. You can choose to switch this out with Wit's end to deal more damage on-hit, or build it normally for the AP and zoning.

Guinsoo's Rageblade I chose because it works well with your Swicheroo. You already need auto attacks to speed up your auto attacks, so it's not like it was this big requirement standing in your way. You are somewhat AD, so you enjoy both the AD and the AP. It's a great item for improving your dps and thus you get it as soon as you have some sort of on-hit to abuse.

Wit's End grants you more bulk, provides you with on-hit damage and makes you even better at taking down mages. You actually steal magic resist with each auto-attack while also dealing bonus damage with each hit. If it weren't for the fact that Nashor's Tooth provides you with sorely needed AP (as well as attack speed and on-hit to tie it all together) and Guinsoo's didn't lay the foundation of infinite attack speed, it would come earlier. Combined with Guinsoo's Rageblade and Switcheroo, you become deceptively powerful.

Rabadon's Deathcap is pretty self-explanatory. More AP. Always good. It is around this time that you can sell your Doran's Blade to get the other prerequisite for Rabadon's. You'd much rather have a Ridiculously Heavy Rod than a Doran's Blade at that point.

Zhonya's Hourglass evens out your defenses into something strangely tanky (or at least not a glass cannon) while providing more AP (you have been seriously lacking in that department before Rabadon's) and some breathing room if you find yourself caught in a dumb situation.

It's simple. Once you have completed a build, you can tack Alacricity or Homeguard onto your boots because you can. Then you can buy an upgraded totem and some Elixirs because why not, you have the money to burn at that point.

Generally, Alacricity is for when your team is winning, Homeguard is for when your team is losing, but Alacricity does provide more dodging ability either way and Homeguard quicker travel. It more or less comes down to preference.

Void Staff is there if you are on top of the world and have absolutely no need for the self-preservation Zhonya's Hourglass provides, though it kind of clashes with Sorcerer's Shoes and Wit's End. Still, it's an option.

Guardian Angel provides more protection in exchange for AP. The situations where this is more useful than Zhonya's Hourglass are few, but exist.

Furor seems like a common-sense choice when so much already depends on you attacking something, however, there will be times when there are no minions in sight, (or when you need to just run) and with your Super Mega Death Rocket, Zap, Flame Chompers, Switcheroo, Ignite, Flash, Basically your entire kit, it is unlikely to be the deciding factor of whether or not your opponent gets away.

Edit: At this time, it provides less movement speed than Alacrity, has an activation requirement, and doesn't stack with itself. Possibly a tooltip error, but I wouldn't chance it.

Switcheroo! Gives you the highest innate attack speed in the game. This is what makes APAAC Jinx a thing and this is why you build on-hit items rather than pure AP. It is also very helpful for holding your own in lane and zoning successfully. It's flaw, its slow start-up, is practically mitigated by all your attack speed. It offers you so much dps that your zoning might as well be a ruse. Max it first.

Zap! Is usually what you get next, but feel free to switch it out with Flame Chompers (picking it up at level 4) depending on your opponent. Early game, you are essentially AD, so don't worry about relying on Flame Chompers. Use Zap to control the gap and keep your enemies away. Since you are AP, always max this last.

Flame Chompers! Is what you really love to zone with. throw it in people's faces and use it to keep yourself safe. If they get caught, give them a Zap or a missile. Later on in the game, these can really hurt. Toss these into team fights and they can really do some damage. You will be using these far more than an AD Jinx. Max this second. Now, you may be wondering why I choose to max Flame Chompers second when I'm building AP. Well, why is AP Jinx bad? "One skill that scales off of AP isn't worth it alone" is pretty much true; Flame Chompers, even with it's good AP scaling won't win you the world. As such, I focus more on the on-hit potential that Switcheroo offers.

Super Mega Death Rocket! Is your ult, and even if it scales off of AD, it is still your ult and still does a nice amount of damage. It also scales off of your opponent's missing HP, so there's that.

AP Jinx is rather self-explanatory. Try it, you might like it.