Gangplank is a rather easy to use champion. He has average health, strong attack power, and average spells. This guide will attempt to exploit his strong attack power by giving him high critical strike chance and high attack damage. He also isn't very mana dependent, especially late game. You might have some problems with mana early game if you spam too much.
Overall, Gangplank is very fun to play and is one of my favorite champions.

Pros:
Great burst damage late game
Has a crowd control heal
Area of effect ultimate
Ult can be cast anywhere on map


Cons:
Squishy
No reliable crowd control
Item dependent
Only two skills do damage to enemies
Long cooldown for ultimate

For the mastery tree, I go 21-9-0; this is pretty self-explanatory. We're going for good burst damage, so offence masteries are what's needed. We don't really need any of the low level utility masteries, so we go with 9 in defense because our Gangplank is pretty squishy with this build.

For our runes, we use 9 Greater Mark Of Desolation and 3 Greater Quintessence Of Desolation to make our attacks do more true damage. Our 9 Greater Seal of Critical Damage give us a boost on our Critical Damage, which will make our critical hits do more damage. This will help immensely late game, when our critical strike is at or near 100%. Last but not least, we have 9 Greater Glyphs of Celerity Greater Glyph of Scaling Cooldown Reduction, to lessen the burden of our super long ultimate cooldown.

Starter Items
For your starter items, buy the Boots, 2 Health Potion, and a Mana Potion. This will give you some speed and allow you to stay in the lane longer in the early game.

Early Game Items
Once you have enough money, recall and buy Zeal, and eventually build up to a Phantom Dancer. These give 55 attack speed, 30% critical strike chance, and 15 movement speed each. Next save up and buy an Executioner's Calling. This gives 18% lifesteal and 15% critical chance, so you won't be as squishy.


Mid Game Items
Now you're going to buy your second Executioner's Calling. Next start to build up to the Infinity Edge. To build up to it, buy the Cloak of Agility first. Ideally you want to buy the B.F. Sword next, but if you find yourself at your base with enough for the Pickaxe but not the B.F. Sword, buy the Pickaxe first. Now you'll be hitting criticals every time and should be doing some really good damage.

Late Game Items
The game will now be coming to an end, and your team should be taking out the Nexus turrets. To seal the deal, sell your boots and buy the Frozen Mallet. This gives 20 attack damage and 700 health, allowing you to lane longer and hopefully keep you from getting nuked too badly. To build up to it, buy the Giant's Belt first.

An alternate to the Frozen Mallet is the Bloodthirster. This gives 60 attack damage and 15% lifesteal, and its passive gives one extra physical damage and 0.25% lifesteal per kill, up to a maximum of 40 extra physical damage and 10% extra life steal. This will help you lane longer and end the game.

For my Summoner Spells, I usually use Exhaust and Flash. Exhaust makes it easy to nuke people down, and Flash makes for pretty easy getaways if you know the right spots to use it.

You could exchange Exhaust for Ignite, because often times champions seem to get away with low health. Flash could also be exchanged for Ghost, it's really just a matter of which you like more. The downfall of Ghost is usually that people can put some type of CC on you before you get too far away, but with Gangplank you have your Heal Scurvy, so that's not really a problem.

This is a very helpful passive. Grog Soaked Blade applies a poison that deals damage per second, increasing per level, and decreases incoming healing by 50% for 10 seconds. This helps immensely when you're chasing a low health opponent, and often means you don't have to turret dive. Just keep this passive in mind when you're chasing someone into a turret.

Parrrley is Gangplank's only damage dealing ability, other than his ultimate. However, it's a very effective one. He shoots a unit for 20/ 55/ 90/ 120/ 150 plus 100% of his attack damage. This makes it very worthwhile to get some good attack damage. Parrrley can also crit, so with 100% critical chance and 250% critical damage, the ability can do over 1000 damage late game, and only has like a five second cooldown. As an added bonus, Parrrley gives Gangplank 4/ 7/ 10/ 13/ 16 extra gold if the ability kills its target, so it's a good idea to last hit minions with it.

With Remove Scurvy, Gangplank consumes a large quantity of citrus fruit, clearing any crowd control effects and heals him for 70/ 125/ 195/ 280/ 380, plus 100% of Gangplank's ability power. This can be used to help you lane longer and will also help in getaways. In an ap Gangplank build, this ability can also heal Gangplank significantly.

Raise Morale's passive increases Gangplank's attack damage by 8/ 10/ 12/ 14/ 16 and his movement speed by 3/ 4/ 5/ 6/ 7%. It's active increases Gangplank's attack damage by 20/ 28/ 36/ 44/ 52 and movement speed by 14/18/22/26/30%. It also increases allies' attack damage by 10/ 14/ 18/ 22/ 26 and movement speed by 7/ 9/ 11/ 13/ 15%, this lasts 6 seconds. Raise Morale is great for chasing and fleeing.

In my opinion, Cannon Barrage is one of the most versatile ultimates in the game. It can be cast anywhere and used in numerous situations from teamfights to escaping to picking up kills on low health fleeing champions. For 8 seconds, Gangplank's ship fires cannonballs to an area, slowing hit enemy champions by 25% and dealing 65/ 110/ 155 +0.2% of ability power per hit. There are two drawbacks to Cannon Barrage: It has a pretty long cooldown, about 2 minutes all game, and it isn't always very accurate; how many times you hit the enemy is just based on luck.

Prelude:
First of all, I don't recommend soloing with this Gangplank build, or with any build really. Second, lane with a tank. In my experience, Sion is a great laning buddy for Gankplank; but anyone with a good amount of health and a slow/ stun will suffice. You want to let your tank initiate fights and be focused. You will be behind him, spamming your Parrrley and Raise Morale. If you get focused, you're probably dead unless you can get off your Exhaust/ Flash combo.
Be careful also to not spam your Parrrley too much, or you'll find yourself without much mana and needing to recall often.

Very Early Game
Until level 6, you really can't be too aggressive. Use your Remove Scurvy to keep yourself in the lane and farm minions. When you're ready to get a kill or two, you want to harass your opponents to around 1/4 health with Parrrley, then use Raise Morale and your buddy can slow/ stun them and you will get the job done with Parrrley and normal attacks.

Early-Mid Game
Now that you have your ultimate, Cannon Barrage, you can afford to be more aggressive. If you find that you've pushed too far too fast and are being chased, place your ult on yourself and safely retreat to your turret.
You can also help your teammates with Cannon Barrage. If there's a fight going on somewhere and your team needs some assistance, take advantage of your ult's range and place it on the enemy's champions in the fight. You can also use it to slow/ finish off fleeing enemy champions. If they're recalling with really low health at a turret, call in a barrage on the turret and you just might get a kill.
Once you have your 2 Phantom Dancers, your really going to start doing work with Parrrley. You probably still won't be able to 1v1 anyone yet, but 2v2s and even 2v3s should be a breeze.

Mid-Late Game
Once you get your two Executioner's Calling, you should be able to effectively gank other lanes. Just use Raise Morale to speed over to the lane, call in your Cannon Barrage and relieve your teammates of opposing pushers.
You'll also now be a prominent force in teamfights and be able to hold your own in 1v1s. If your team focuses champions one by one with your Cannon Barrage on the fight, you should win almost every fight.
In a 1v1 situation, use Raise Moral to do some good damage with normal attacks. Don't use Cannon Barrage do early in the fight or it won't be as effective. Use it when your opponent starts to run, then finish them off with Parrrleys.

Late Game
Now you're at 100% critical strike, and you should be able to easily 2 man baron with a tank. You should be able to lane for long periods of time, if you've started to build up to the Frozen Mallet. Pushing should be no problem, and neither should super minions, if it comes down to a close game.

Well, that wraps up my crit build for gangplank. This is my first build made here on MobaFire, so don't be too harsh. :P Gangplank is a very fun champion to play, and I encourage you to try out my build. Please post your thoughts, errors I've made, and how you think I can improve the build. Thanks.