Tactics on how to siege a castle in Shogun 2?

Tactics will vary depending on your army composition, and that of the enemy. I'll mostly be talking about PvE, as that is my strong point, having played only a few field battles in the multiplayer portion of the game. IIRC, in PvP you only get 40% of the attacker's army size for balance purposes: It is obviously very advantageous to defend. Luckily, the AI can be beat with an equal-sized army on a regular basis, and with even less sometimes.

I always run a purely ashigaru force on Legendary, rarely bringing cannons "for the lulz", and almost never using matchlocks in my army composition. Hence, my advice on exotic stuff is limited: Siege weapons seem mostly useless, forcing you to take on a repair bill as well as a public order decrease, having damage that's mostly inferior to a normal ranged unit with the only benefit being range, and IIRC slowing your army down. Guns are great as a defender and useless as an attacker. Stealth looks neat, but is mostly pointless.

Bows are pointless in a siege, especially as the attacker, if you can't get a full quiver's mileage out of them - they just can't do the damage of their melee counterparts if you only get five or six volleys in, and they cost more. If, for any reason, you can't get ten minutes of continuous arrow fire to use up all your ammo, if you have to make a charge for the walls under enemy fire, use you bows as cannon fodder. Move them in range of the enemy archers before your melee units, preferably in a scattered formation and as a very wide, shallow line. The same should be done if you have to scale the walls under fire and have already used up your bows' ammo (great job!). They don't necessarily have to just stand there either: You can use them to scale the walls first, often on the side opposite to your melee units, sacrificing them in droves to force enemy units off your main approach route. This will often get enemy archers to leave the walls, turn around, and shoot at your diversion, making the main wave's push 100% safe. On the very rare case where they don't, you can still engage them yourself. Still, scaling walls bow-first is a bloody endeavor, and should be avoided unless you really know your stuff - it's a very good way to waste five to ten units for a marginal benefit that wasn't necessary. Burning down the gate and charging in a clumped mass of 600 bodies should give your diversion a ton of staying power, but for the most part, keep to just using scattered lines to absorb enemy fire.

Now, I advised using up all your arrows. How do you do that, you ask? Flanking. Unless you're fighting a 20v20 siege with a good amount of archers thrown in, your enemy will have an unguarded portion of the wall. The AI tends to stack archers against your melee units and your general: positioning your archers on the opposite side of the fort, or on the flanks, will usually get them facing unguarded sections. As long as you keep your melee units in place, the enemy bows will obligingly stand and watch in the only direction they can't fire in, waiting for those 500 yaris that are just outside arrow range (or 1km away, it doesn't seem to matter) to close in, as you gleefully slaughter their comrades. If any towers are shooting at your archers, you can burn them down by using melee mode, which gets your archers to chuck torches at them just like a melee unit: You don't need to have fire arrows researched, and even if you do, save the ability to boost damage and don't waste it on structures.

In this screenshot, four bows are facing the three yari on the opposite side of the fort, offscreen, and two are facing the direction my own bows and the general came from initially. They never moved from there and were later killed in melee. My own bows are not at max squad size due to previous battles; very few were killed as they approached through the forest (which acts as heavy cover against arrows), and not a single arrow was fired at them after they got in position.

What units to target: Generally, focus on melee units first, cavalry second (unless you're using a ton of katanas and no spears), and don't even bother with archers, even if they are out of cover. Get squads down to 20%-40% of their maximum size and move on to another: shooting at a unit of 20 soldiers is a waste of ammo. Shoot on units that won't change their direction until the arrows land: Your archers will lead the enemy, but they don't do any advance prediction, and always assume that the target will keep its speed and heading for the projectile's entire flight...even if there's a castle wall a meter ahead. Shoot either at units standing still, or at units moving in a straight line that are unlikely to stop or turn soon. Disable manual fire. For best effect, shoot at the flanks or rear of enemy formations. Target deep formations relative to your shooting direction: This way, even arrows that miss have a good chance of hitting the troops behind. Sneak in as many generals as you can in your group of archers: Sometimes, they'll get enemy archers to reposition, in which case you should return them to your melee units. When this doesn't happen, the inspires they provide are great for boosting arrow efficiency, while Stand and Fight makes everything better.

If the enemy does have enough bows to cover the entire wall, don't panic. It's fully possible to burn down a gate and sacrifice a single Yari Ashigaru per Bow Ashigaru troop (if you're less confident in yourself, go for three units for two troops) to make an opening. They'll be swarmed and die, but provided you time it reasonably well, they'll destroy all but a dozen of a previously full squad. Against samurai and above, you're probably out of luck: Unlike ashigaru, they can survive reasonably well in melee, even against expensiver units. Still, except for Kyoto, most sieges don't involve a 100% samurai defense. If enemy bows move in to intercept your flanking force, peel off a squad and abuse another opening, gradually adding more until all enemy archers are stuck waiting for a single bow unit (preferably depleted) to come in range, in a situation analogous to those camping the meleers.

Don't tire your men. The main (only, even) reason the AI can't to reliably win sieges with twice the defenders' force is because it can't stop running. Running is not needed in a siege: Even the 10-meter deadzone before under the walls can be walked through, although saving energy from that is needlessly risky. Once your melee units have scaled the wall, they are every bit as capable as the enemy, only handicap being the amount of fatigue they've accumulated from the climb: Hence, keep to scaling short sections if you can help it. Scaling is best done from every direction, to flank the enemy: While morale doesn't matter, a troop in the rear inflicts many more kills, decreasing your own attrition. For the melee phase of the battle, don't forget to use your depleted archers: having 1200 men stand around is a huge waste. The most basic thing they can be tasked with is capturing the fort control zone: a mass of even cheap units can easily crowd the place and make it unreachable to enemies unless they can cut at a heroic pace. The only problem is getting there in the first place, easily solved by scaling second, as a flanking force, from multiple directions. Archers can also be used to tie down enemy bows or charge flanks. Be bold when using them and prepare to take huge losses with little gains: Soon, you'll learn what works and what doesn't.

Scaling the walls is generally for when you're wiping the enemy out, burning the gates down is for when the battle isn't going your way. When scaling, the enemy should only have a third to a tenth of their original melee force left (and who cares about archers, they suck at melee and invariably get forced into it within the first fifteen seconds of the charge), allowing you to swarm and massacre. If there are a lot of enemies left, presumably because you failed to empty your quivers or because you attacked with a much smaller force (good job, but why are you reading this then?), I'll describe a moderately advanced trick I try every once in a while that may work for you. Burn down a gate and make a deep yari wall formation in guard mode get inside, but don't expose their flanks: there is a castle model where some of the gates have inner walls, allowing the entire formation to get in while only exposing the front. For most, you just have to support them from the side or barely get inside. A yari wall has great staying power, especially if you manage to get enemy yari to attack it first, as opposed to katana units. If all goes well, soon the entire garrison will be swarming your diversion: they can easily hold out if the flank is safe. If not all units are engaged, burn down a second gate and approach with a similar formation, tying up the rest. Then, scale from a third direction using a squad with big unit size, or even several, massing the fort capture point: put a wall of bodies in front of the area, making sure that the enemy can't kill enough in a minute to get to the location. Your bow squads are perfect for this. Usually, only one or two squads will break off to meet you, letting you capture the point with little losses against what could have been unwinnable odds.

Random bit of info: An unupgraded, unleveled unit of bow ashigaru without access to inspire has just enough ammo to kill the free samurai retainer unit.

PS: Army composition: The more archers you have, the harder it is to utilize them properly. 16 melee 3 archers is reliable but unremarkable. 16 archers 3 melee is hard to pull off, and is still reliable (this time, in bringing even the best players some embarrassing losses), but can net you amazing kill:death ratios against superior forces. Stay in the 7:12 - 12:7 range for a good mix of reliability and efficiency.