First I want to say that I know this build has some flaws, but this is the way I play Gangplank right now. If I had more IP I would probably get different quints, and maybe MR per level glyphs, but this is what I am working with now. Other things like, whether or not to get a philosopher's stone after sheen or what kind of boots to get if ninja tabi wouldn't be effective are open for adjustment depending on the circumstances. The main idea is to build Gangplank tanky after you get sheen, boots, and Infinity Edge.

The purpose of this build is to be able to take out enemy carries in team fights. This build allows you to take a good amount of punishment, while still dealing good damage to squishies.

I have had a lot of fun playing this build, because with this build, you don't have to worry as much about getting instantly killed. You can run through the middle of a team fight and get in the face of the squishy of your choice.

I choose exhaust over ghost because Gangplank is already very fast, and has Raise Morale making him even faster. Also exhaust will shut down one of your main targets, AD carries.

Flash is very useful on Gangplank, because it can help you reach squishies easily, but more importantly, it enables you to escape if you are being heavily focused, along with Remove Scurvy. Although you are a lot tankier than normal, you are still no Amumu or Rammus. Use flash to get out of harms way if you are near death in a team fight. Then hang around on the outskirts, healing up with Remove Scurvy, and buffing your allies with Raise Morale, until you are ready to jump back in the fight or chase remaining enemies.

Within 25 mins, you should be able to get your sheen, lvl 2 boots, and IE (and philosopher's stone if you choose to get one). For the early game, you should be playing Gangplank normally. Although you have your defensive runes and some masteries, you aren't tanky enough yet to run carefree through the enemy team. After you get your core damage items, get pieces of your tanky items. I usually get a giant's belt, and then armor or MR depending on the enemy team composition.
Towards late game, your items should be,

Boots
Sheen
IE
Warmogs
Atmas
FoN

After this, I typically sell sheen to get more armor or MR, making Warmogs more effective. However, it is your choice. You could sell sheen to get a better damage item, possibly get a frozen mallet, or upgrade sheen to trinity force.

I use this build in normal games, to counter the glass cannon carries that are so popular. Although this may not be the "best" Gangplank build, it is very fun. I am a tank player so I am used to being able to sit in the fray of battle and mess with carries. I am not very good with glass cannon DPS and I have never really had fun playing them, because you have to play so cautiously and carefully. I enjoy this build a lot because in team fights, you really can just run through the enemy ranks and get to the high value targets. With Gangplank's tankiness, Flash, speed (with Raise Morale), CC removal/heal with Remove Scurvy, he can be very hard to kill.

Try the build out and see if you like it.