Not Updated For Current Season
Tankplank is the way I feel gangplank should be played, as he is melee range, thus making him need more armor and resists than a ranged carry, while still doing insane high damage crits and helping his team with slows.
Early game farm/deny/lane control is important with tankplank. Start out with your Doran's Sheild (if Solo lane) or mana crystal+2 health pots if in a duo lane. Before minions spawn wait at the nexus. As soon as creeps being to flow out, use Raise Morale to kill 2 minions off by the time they make it to your lane, walk with them as you wait on cooldowns.
Early levels ( 1-4) are extremely rough on mana. Use your mana to kill off creeps rather than parrly'ing creeps. Deny every cannon minion in the lane no matter your current state of mana. This should give you an early 1 level advantage.
Medium-early game (5-9) is where you begin control of your lane. Mana is not as bad now thanks to runes and you can begin your intense farming. Start Parrrlying, order of importance is Cannon Creep>Squishie Champion without heal/regen>regular creep>more durable champion. Your runes will allow parrrly to hit a few hundred damage on crits which is enough to keep the lower level opponents at bay. This is gangplanks time to shine. Abuse the time when you are level 6 and they are just hitting level 5, and are probably parrly'd down to half health to Raise Morale>Ult>Exhaust>Attack>Parrrly straggler. You can probably score a kill with your crit runes and ultimate.
Bluepill asap to buy catalyst/heart of gold. These are your lane priorities and you will want to hold them as long as you are leveling and farming the early game.
Obtain up to Merc. Treads before midgame, and you are on track.
Midgame (level 10-15) Continue to push/farm/kill/deny as much as possible. Drag out lane phase if you can and try to establish a huge significant level advantage as well as gold advantage. Zone with your awesome damage on parrrly crits and phage, and continue to deny your lane foes of gold and exp. By this time you need to finish farming Randuins, which when combined with your ultimate provides a HUGE teamwide slow and potential for damage, making him a great initiator with remove scurvy and the combination of R and Randuin Active.
Lategame (Level 16-18-end of game) This is where tankplank comes fullcircle as an amazing champion. While still providing huge utility with randuins, parrrly slows, and Ultimate slows, Tankplank's damage amplifies immensely once Atma's Impaler is bought. Only to be followed by the luxurious IE to top off his damage. This combination makes you strong enough to kill many many champions.
In Teamfights: Raise morale a creep for the team buff, Ult, pop Randuins active in your ult, Apply slows with auto attacks and parrrly, chase and destroy the other teams squishie casters or carries, win the teamfight untouched and flawless.
NOTE- If you are being harrassed hard in lane, drop points from raise morale and place them in to remove scurvy and begin to heal more often. Warning: this does lower farm rate, deny rate, and reduces mana over time.