If you don't know why you would ever want to play AP Xin read on if you already know the introduction is not that important.
AP Xin is all about his passive this will heal him every third basic attack. This heal has both an AD and AP scaling 10% and 40% respectively.

Pros:
High jungle sustain.
Tanks are never going to kill you as long as you are basic attacking them.
Tanky enough not the get deleted
Reasonable dps.

Cons:
Bad duels early game.
Hard CC stop you from healing.

Bloodrazor: is gonna help you in damage throughout the entire game, as Xin does not have a lot of AP damage scalings he needs it(and it's obviously attack speed is gonna help you).

Rageblade: This one is obvious, larger more frequent heals and a damage boost on your basic attacks. The double on hit every second hit with make you heal every second aa instead of every third.

Spirit Visage: Increased healing and more survivability. The cdr doesn't hurt either since your Q will cut the cooldown of your other abilities by 3s each time it is used.

Zhonias: A source of both armor and AP, the active is also very useful since it can evade a cc chain or a large burst of damage that would otherwise kill you. You can also use it to wait for your E to come back up if your are getting kited while it is on cooldown.

Nashors Tooth: More attack speed and AP. This items grants you second most healing/sec after rageblade. The passive also makes you deal significantly more damage.

Phantom Dancer: Reduced damage from a single enemy. Strong in a 1v1 or while your ult is on.