Gangplank is currently an extremely popular champion. I've played him since before the rework to his passive. The main build in this guide is aimed at success in solo queue. it has enough survivability that you wont be instantly bursted down by assassins and burst casters while maintaining enough damage that you should be able to carry your team to victory.
This guide is still a work in progress so go easy on me :)

Quints: I find Attack Damage quints to be effective, it gives Gangplank suprising burst at level 2 when raise morale is ranked up. Other options include Armor Pen and Critical strike chance.

Marks: Armor Penetration is pretty much standard for any champ that scales with AD. I dont recomend using anything other than armor pen for marks

Seals: I find armor particularly useful especially given in terms of ranked you will probably facing another bruiser such as Irelia or Renekton. Other options include crit chance and attack speed.

Glyphs: Lately Ive decided that attack speed is a good option. Attack speed scales pretty well and is good for gangplank because it allows you to stack your passive as quickly as possible. Other options are magic resist, magic resist per level and crit chance.

As you can see Ive put crit chance as an option for most categories. Some people like to run a lot of crit chance to get early burst from parrley by getting more crits.

Pretty standard. You could actually run the magic penetration mastery to help your ult. Mana regen instead of the greed is fairly common also.

Executioners calling is a particularly unique choice. I get this first because it provides crit to give some nice early burst from parrley crits well also giving nice lane sustain. This could be sold for a Bloodthirster once you reach full build. An alternative to this would be to run the crit chance runes mentioned earlier and then start with wriggles lantern, which provides a lot of more benefits but no crit chance.

Merc treads are generally the best boots to get, however if they have low cc/magic damage, then ionian boots and beserkers greaves are also options.

sheen is one of the most important parts of the build, if you get fed and get that sheen early you can snowball because of the sheer damage it adds to parrley. Easily one of the strongest early game items for Gangplank. I like to just build that into trin force right away, which gives some more crit and damage from zeal/phage.

From there if you're going well you may want to build directly into Infinity Edge. Infinity and Trinity force togethor will mean that you will literally kill people in 2-3 parrley crits if they havent built any armor.

Finally atmas and banshees veil, both add decent surviability to gangplank while also giving him damage thanks to atmas passive. Of course atmas gives you crit chance so you will be putting out your crits more often.

Of course remember to always build situationally too. If the opposing team has 4-5 AD champs including 2-3 carries, it may be more benefical to build thornmail or frozen heart than atmas.
Not to mention if you're facing suppression or someone like mordekaiser then you may want to build Quicksilver sash instead of Banshee's veil, just to make sure. (Keep in mind that remove scurvy also breaks supression)

In certain situations it is best so start with a philo stone rather than executioners calling. I do this when my team has a jungler and the other team doesn't and I have to hold a lane 1 v 2 or if I'm against a strong harrass opponent.

Exhaust- Easily the best spell for an AD champ. If you find yourself jumped by another AD champ exhaust will let you either kill them or escape.

Flash-Easily the best summoner spell. Allows you to escape from sticky situations and as you get used to using it you will learn to use it offensively as well.

Other viable spells include Ignite, Teleport and ghost.

Spells I dont recomend:

Clarity-Ive seen some people use it and sometimes you run oom if you're spaming left right and centre, harrassing and using parrley on every minion, but in my opinion its unnecessary and just taking up a slot.

Heal- Just no, the best this might get you is an early kill because your opponent is so surpised that someone is acutally using it.

Revive- Only ever really viable on karthus and old tf/pantheon.

Rally- I dont even wanna...

cleanse- you could use this but I think remove scurvy and banshee's or qss is enough to protect you.

Fortify- Viable only for some tanks and supports but even then, it's rarely used.

Pros
-High damage output
-Ability that breaks cc
-Can carry bad players
-Good suviability
-Good base stats and late game scaling
-Global ulti provides great support


Cons
-Can be focused prior to building survivability items.
-Will be called OP in normal games/low elo.

Make sure to focus on last hitting as much as possible early game. If you throw 1 or 2 parrleys at your opponent(s) early they may be hesitant to go in you again, allowing you to farm free from harrass. If you're good at last hitting this will be extremely advantageous.
During mid game you will probably spend a lot of time doing what is known as afk farming while watching for oppurtunities to use your ult to assist your team mates.

When late game comes you will probably be able to steam roll in fights. If you get focused then your ap carry and ranged ad champs should clean up the fight and if you dont get focused then you should be able to clean up the fight.

Gangplank is a great champ to play, contrary to some people's beliefs he is neither OP nor is he boring, however he can be hard to deal with for players who are not very skilled at this game, almost on the level of various assassin champions.