You've got some options for starting items. Take
Cull in anticipation of less early game harass.
Doran's Blade is for an average match up, and
Doran's Shield is a more defensive option.
Sapphire Crystal builds into one of your first items and gives the mana to spam
Parrrley. If you've got gold leftover, take one or two
Health Potions, and don't forget your
Stealth Ward.
Sheen is all you should need for early game minion farming: A stronger
Parrrley, mana, and 10% CDR. Use
Parrrley to start raking in those Silver Serpents. Also, buy a vision ward to help prevent ganks, and buy replacements when it is destroyed.
Your first complete item is
Essence Reaver It provides CDR, crit chance, and AD to further empower your parleys. But the best part is the new passive effect: "Critical strikes restore 3% of your maximum mana." And your
Parrrleys can critically strike, so with this item, it is possible for you to gain back mana from using the ability! I have tested this, and it does work. You really do get mana back

The Math
If you have 100% crit chance, you will gain 3% of your mana with every
Parrrley. You need to have 1334 maximum mana in order to gain 40 mana with each crit. We have close to these stats once we reach LV 18 and full build. And remember, your basic attacks will also restore mana if they crit, so building unnecessary mana isn't really worth it, and don't be too worried if you can't reach 100% crit chance.
Feel free to spam
Parrrley whenever it is off cooldown and can kill a minion. Combine with
Powder Keg to farm multiple minions at once. And with only 800 gold, you can buy
Cloak of Agility, which will provide an additional 20% critical strike chance and tops off your CDR at 40% thanks to Essence Reaver's other passive effect "Gains up to +20% additional cooldown reduction equal to your critical strike chance from other sources" (prepare for this to be nerfed in the future; it's way too OP). Again, I have tested this, and you really do get 30% CDR from that item.
Add
Infinity Edge,
Phantom Dancer, and
Trinity Force into the mix to have near 100% critical strike chance and tons of damage. If things aren't going your way, you'll need to go tanky too.
Dead Man's Plate provides even more damage to your first
Parrrley, so it's a natural fit.
Spirit Visage allows you to Eat Oranges and be extra K. And
Sterak's Gage will help you avoid burst and stay in the fight longer.
Gangplank used to have trouble building core items with CDR, forcing him to buy
Ionian Boots of Lucidity. Now, we reach 40% CDR with ease, so GP is allowed to freely choose his footwear. As for enchantments,
Enchantment: Distortion lowers the cooldown of his Summoners buy 20%, meaning that they are available more often to make plays.
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