First off, I would like to make it clear that this is not a joke build, AP Tryndamere is just as viable as the traditional crit based DPS Tryn. AP Tryn does not have the same level of sustained damage as DPS Tryn, but instead functions as a hit-and-run character with insane survivability. AP Tryndamere is also much harder to counter, since standard anti-carry counter such as Thornmail and Randuin's Omen are ineffective due to him not being dependent on his auto-attack like most carries are.
The rune choice for AP Tryndamere is simple; armor pen is the only viable Mark and cooldown reduction is the only viable Glyph. Flat armor Seals are helpful to increase his survivability early game, and movement speed Quints are essential for chasing and escaping.
There are two good ways to build AP Tryndamere's masteries: 1/21/8 and 21/0/9. 1/21/8 gives him great survivability and a small late game AP boost, as well as the Exhaust and experience masteries. 21/0/9 increases your attack, which is still useful early game for last hitting minions and getting kills at low levels, but it grants him much less survivability than 1/21/8. It's also noteworthy that the armor pen mastery benefits Spinning Slash, which is your main source of damage.
These three items are the core of your build. AP Tryndamere really starts to shine upon completion of Rabadon's Deathcap. To all those wondering why it's a good idea to rush DFG, its damage is insane early game, since most enemies won't have much magic resist yet. The DFG + Spinning Slash combo is amazing, especially if you have a lane partner with a snare. Rabadon's Deathcap makes DFG, Spinning Slash, and Bloodlust's heal so much more powerful, let's you dominate team fights.
If you're not doing so great after you get Rabadon's Deathcap, skip the Mejai's Soulstealer and go straight for Lich Bane. However, if there's any perfect time for Mejai's it's right after Rabadon's, because the 200ish AP from Rabadon's will be sufficient until you can get a bunch of Mejai stacks. If you chose to skip Mejai's, fill in the extra slot at the end with another Hextech Gunblade or Deathcap, or a defensive item like Guardian Angel or Warmog's Armor.
Lich Bane is a simply amazing item on AP Tryndamere. It gives him 7% movespeed and a bit of AP and magic resist, but most importantly, it makes your auto-attacks roflpwn enemies. Your damage output will be insanely good as long as you auto-attack between using abilities.
Hextech Gunblade is a nice item with both AD and AP and a ton of lifesteal and spell vamp. It also comes with a range slow that does reasonable magic damage. Hextech Gunblade is not vital to this build, however, and can be replaced with extra Rabadon's Deathcaps.
Get a point in E, then Q, then W, then R>E>Q>W.
Spinning Slash scales 1:1 off AP, and pwns in teamfights mid-late game. It has only a 9 second cooldown at level 5, and it becomes even lower with your insane CDR, and serves as a great gap-closer and escape ability.
Bloodlust is give you damage but we don't really care about that. It does, however, give you an amazing heal that scales 1.5:1 off AP. This means that with level 5 Bloodlust, you can heal for 770 if you have 500 AP, with a only 12 second base cooldown (although you'll have almost 40% CDR).
Mocking Shout is a great ability that makes chasing enemies a joke. However, it does not scale off AP and only one point is necessary for you to chase enemies down.
Undying Rage is just as OP as it is on DPS Tryn. If you use Bloodlust, then Undying Rage two seconds later, Bloodlust will recharge in time for another uberheal, making it essential for escaping and drawn-out battles.
Exhaust and Ghost work great for chasing enemies and escaping. Ignite and Flash could also work well, it's really a matter of personal preference.
TLDR: Rush Ionian Boots, DFG, and Deathcap, and start pwning people with DFG and Spinning Slash. Don't be too aggressive until you get DFG, and as long as you play carefully and don't try to 1v1 their carries or mages, you'll be a monster in teamfights and you'll carry your team to victory.