Gangplank bot is very fun to play. In most matchups you can farm comfortable and reach your power spike without being abused or zoned. This gangplank playstyle is not recomended for SoloQ or DuoQ games. It is just a different way to play botlane just for fun.

Pros:

High oneshot damage

Fun to play

Viable

Unexpected pick


Cons:

Requires skill to farm

Strugles with mana early game

Must duo with supp

Hard counters

Primary:

Always go for First Strike. It provides with extra oneshot damage and extra gold, so you could reach your power spike earlier. Then always take Magical Footwear for extra movement speed and for saving 300 gold, Biscuit Delivery to minimize mana problems in early game and Time Warp Tonic to surprise your enemies healing suddenly a little and turning back a trade or fight.

Secondary:

If the enemy play a botlane with high early damage or with high AA range like Lucian, Caitlyn, Draven... go for Bone Plating and Revitalize for extra sustain and survivability. If it is not the case, go for Presence of Mind and Coup de Grace for mana sustain and extra oneshot damage.

In almost any scenario, you want to start with Doran's Shield for sustain against enemy´s poke. Doran's Shield

Your first ideal back is al least at 1100 gold, so you can afford Tear of the Goddess and Sheen. If you don´t have enough gold, prioritize Sheen.

Your first item must be always Essence Reaver as it provides lot of damage but also ends Gangplank´s mana issues. At this moment, you can farm with Q and E+Q as much as you want.

After that, focus on building Prowler's Claw and Manamune. Build first Manamune if you have Tear of the Goddess with over 200 stacks. Build Prowler's Claw first if it´s not.

Then go for the rest of the build. If the enemies are mostly squishes, buy The Collector and finally Infinity Edge. If they are more tanky, buy Lord Dominik's Regards and Infinity Edge. If the enemy team is 50/50 squishy/tanky, buy The Collector and Lord Dominik's Regards.

You can also change any of this last items for Serpent's Fang if they have lot of shields.

Gangplank bot goal is simple: gain as much gold as possible. Of course you can all-in if you feel comfortable and you think you can win. But in most matchups this is not even an option. The best way you can gain health advantage and set up a engage is placing a E Powder Keg in a bush and when the enemy aproches, place a new E and quickly Q Parrrley the bush E. If the enemy has not high AA range like Caitlyn, they can not remove your kegs without being expose to damage or engage by your support. So they will have to choose between take the risk or play safe losing farm.

There are few tips to get the best of this Gangplank playstyle:
First, do not feeze the wave next to your tower and do not let the enemy freeze the wave next to theirs. You should keep the wave along the bushes.
Second, if your jungler ganks, use your R Cannon Barrage immediately. The slow will set up a more easily gank and you will get always the assist (and maybe the kill)
Third, look at the map, your R is not very useful in lane but it can turn a duel result in your ally´s favor.
Fourth, prepare ambushes in lane bushes or in the river bush. You can delete or send back to base the enemy adc or support.

And finally, have fun.