Dealing with player character deaths

Rules and expectations

The first thing to do when starting a campaign is to set the rules and expectations of the group you're playing with. Listen to each member and what they expect in terms of their chances of death. There's no point playing a game with a high chance of death if none of your players are willing to lose their characters at any given moment.
Fun comes first and making sure the expectations and rules around death are set will help ensure everybody enjoys the game no matter what happens.

If there are conflicting opinions about PC deaths within your group you'll have to compromise somehow, which could be in the form of meeting in the middle or, if push comes to shove, going separate ways and changing the group. This is why it's important to set the rules at the start of the campaign when people aren't heavily invested yet and dropping out of a group isn't hugely impactful, depending on who you play with of course.

Meaning of death

What does the death of a PC mean? I don't mean this in a philosophical way of course, I mean it in a pretty literal sense. If a PC dies, what does that death say about the player? Does it mean they were just very unlucky or did they perhaps take too many chances? Perhaps they ignored the signs of dangers and ventured into a throne hall, called the king a traitor in front of his guards and thought they'd get away with it.

To some extent this is part of setting the rules and expectations. A game with a low chance of death will usually only lead to a death when fighting a boss for example or indeed if a player does something silly. In a high risk game a death is often just the way of the world.
Either way the meaning of a death will usually impact the way you deal with that death in the game and in terms of dealing with the player, but more on this later.

Responsibility

Tied in with the meaning of death is who's responsible for a PC's death. In most cases it's the game master's (GM) responsibility as they're the one in control of the story and the threats thrown at the PCs, but the players could be responsible for their character's death as well, potentially in a deliberate way.
When a player is responsible for their own death the blame can only be put on them and it could lead to less friction between that player and the GM, which shouldn't be an issue if the rules and expectations are met, but might still occur when it turns out people wanted something different from what they initially thought or when people have a bad day.
But using responsibility as part of the rules can help with making sure players are having fun and know what they're getting themselves into. On top of that it can be used as a powerful tool to make a character's death less shocking to the player, which brings me to my next point.

Tell them in advance

If you're unsure about whether a player will handle their character's death well or whether they're indeed ready to lose their character it's incredibly helpful to simply talk to them about their potential impending doom. Allow them to give input on the how and when for however much this is possible and what this means for the rest of their game.
They could already work on a new character so there's no down time for them and their evening isn't spent listening to others play the game or they could play a character already part of the story but currently played by the GM.

By telling the players in advance you also have the opportunity to make their death meaningful and story driven. Plus since the death is announced in advance it means any story threads tied to that character can be altered to a degree so they don't just end in thin air.
Alternatively you could tell several group members their characters might die next session, so if they tell the other players they won't know who will in fact end up losing a character.

Story versus realism

Story is obviously a huge part of most role playing games and realism often goes out the window even within the rules of that story universe. The same could be done for PC deaths.
Dying to a random wolf is no fun, so the chances of that are generally best lowered even if that means fudging the results of dice, but a major antagonist would usually want to kill whoever opposes them, which includes the party. Dying to them is both realistic and easier to digest as a player.

The death of a PC should be meaningful if they're uncommon. Tie it into the story, make it climactic or emotional and allow the other PCs to mourn the loss of one of their own. Arrange for a session or partial session in which the deceased character is given a proper burial or other service suitable for that character and whatever their wishes were in life.
This doesn't just make the death more meaningful, it also makes the world itself richer and it adds to the immersion of the players.

More story

My previous point about dying to a random wolf not being fun ties in heavily into the kind of story you're playing. In high risk campaigns dying to a wolf could definitely be a thing, but the difference between high risk campaigns and low risk campaigns is often the type of character your players play as.
Most games are about telling a story and within these stories the PCs are the main protagonists, the heroes (or sometimes villains) who manage to do the impossible and overcome whatever obstacles stand in their way. Just like how in regular stories a random, meaningless hero death usually makes for a boring story so too can it ruin a game.

The death of a player character also offers a great opportunity for a dramatic and emotional moment. Everything the party has worked for, everything they've gone through together and everything they still wanted to go through ends in this single moment, assuming this player character isn't revived later of course.
So to avoid wasting this opportunity try to think of a few ways you could make a death really memorable. You're the GM after all, you know what they will face next and when there's a decent chance they might die. Think about how you could describe their death at the hands of this enemy, what happens in that exact moment of no return, how does the enemy respond to this victory?

Blazing glory

Something you could do to ease the pain of the player is to allow their character to go out in a blazing glory, to allow them to make a heroic sacrifice or a final push for whatever their goal was in that fight. In real life adrenaline allows people to perform incredible feats of strength, so why not translate that to the game world as well? There are various ways of going about this ranging from giving them a free turn to making them invulnerable or resistant to damage in their last moments or even to give them back any spent spell points or other forms of expendable skills as a way of showing they're using their last energy before dying.
This not only makes for memorable moments, it also gives the player an opportunity to let their character shine for one final moment, to make a difference one last time.

Now what? - Character

So a player's character is dead, now what? What happens to that character? The character was an integral part of the story, a friend to hopefully many and an enemy to probably at least a few. They may have had a family, they may have had people counting on them or people hunting them. Story threads were likely planned for this character and those in the midst of being solved might now never have an ending.

A good, story driven way of dealing with this is by dealing with it the same way a death in real life would be handled, but tailored to your story universe and the potential wishes of the character.
The character might require a burial at sea, their body might need to be returned to their family and close ones may wish to pay their respects through all sorts of ways. Enemies who may have hunted this character will be pleased to hear this character is dead, which in turn might offer them a chance to fight the rest of the now weakened party.
If the character could be resurrected the party will likely have to drop whatever they were doing and get on with this as soon as possible.

Assuming the character is no longer able to be revived or no longer wishes to be revived the party will likely have to search for a new party member, which sometimes means going back to civilization. Any other ongoing issues with a deadline will either need to be given up or a greater risk may be taken by going without a new party member.
If the character's body needs to be transported it may mean carrying a lot more weight and it could be very suspicious looking, both of which bring their own perils.

Of course this is all assuming you're playing a more serious game and the deceased character was actually liked by the group. If the game isn't a serious one or if the character wasn't liked the body may very well be tossed away for the scavengers, but only after it has been looted of course.

Now what? - Player

A deceased character means a player is now likely without a character to play, so what happens to them? In most cases it simply means creating a new character and getting on with the story, but some choose to put restrictions on what the player can create.
A player may be required to start 1 level below the other party members, start with minimum gold, only be allowed to buy starter gear or have to sit out until the group has had an opportunity to either find a new party member or revive the deceased character.

The penalties put upon a player are usually a means of both making things more realistic and making death less of a reset button for when players constantly want to try new characters. If you don't have players who wish to kill their characters at any given moment just so they can create a new one you obviously won't need the penalty for that, but I recommend using either a monetary penalty (character starts with less money) or no penalty at all no matter what the circumstances are.

Say the player starts with a character 1 level below the others or with gear a regular character could buy with the starting money for that level, they will likely still be much weaker than the rest of the party and thus either die more easily again or drag down the challenge of the story for the rest of the party. Either way it's not exactly ideal for the overall enjoyment of the game.
Now let's say the character simply starts with less money, but does have an equal level and only slightly less powerful gear (depending on the gear the party has, most people wouldn't have legendary items for example), in this case the lack of money could be a detriment if the rest of the party isn't willing to pitch in for the cost of supplies or potions for example, but if they are the "punishment" for dying is shared among the party and thus less of a punishment for a single player.

Punishment

Let's delve into the idea of punishing a player for dying a little more. Putting restrictions on what a player can create in terms of a character may not always be intended to be a punishment, but in most cases it will feel like one, especially if the player had no control over whether their actions could've prevented their death or not. Sure, they could've chosen not to attack the dragon, but it's part of the story and you can't exactly flee from all dangers, it'd make for terrible heroes in a bland story.

So by putting restrictions on a player you're essentially saying "I, as the GM, caused your character's death through the way I told the story and perhaps through some unlucky dice rolls and I won't allow you to start a character at the same level as your previous character because it wouldn't be realistic.". Realism goes out the door here, as it does so often when faced with fun versus realism.
The player has lost a character they likely enjoyed playing, that alone is usually punishing enough.

Loot and gear

There are two elements I briefly touched on, which is the "loot" of the deceased character and the gear the new character starts with. First, what should happen to the gear of the deceased character?
The party can usually do whatever they want with it, but if they respected the character they would likely send the character's belongings to the next of kin, if there is any. Any items essential to the story will either be given to the other characters or returned to the next of kin if it doesn't make sense for the other characters to continue whichever quest this item was integral to.

The characters may of course choose to instead use the gear themselves or to sell it, especially since some of the items may be very rare and thus potentially very valuable. However, carrying around extra gear, especially expensive gear can come with its own risks. Thieves and scoundrels may wish to steal these items, either sneakily or through force.
Then there's the issue of actually selling the gear. Nobody will pay the full price of course or else they wouldn't make a profit. But finding somebody who knows how much a rare item is really worth can be quite a task as well, so the players may be faced with a quick and easy, but cheap sell or a laborious and lengthy search for a great and more rewarding sale.

Whenever players choose to keep the gear of the deceased character some GMs choose to deduct this from future loot gained from other encounters, which is fair if the new character starts with equal gear as well. Otherwise you could simply let a character die and keep gaining more and more loot as new characters appear and appear. Assuming they all die without something destroying their gear of course.

End of a story?

When a character dies it may be the end of that character, but it doesn't mean it's the end of their story. Their legacy could live on depending on their deeds in life. Songs might be sung, statues might be erected and tales might shared among folks for years to come. Adding these elements to the story of the campaign can make for some subtle references or even heartwarming moments. An old man might be telling tales of a great paladin, a young child may have a wish to be like that famous soldier or a mother might shows a hope her child grows up to be like that amazing mage. By referencing the deceased character at times you show their story continues and is not forgotten. The players will appreciate it.