A story comes to life when someone tells it for the first time. A story cannot fully exist with just an author, it also needs and audience to be told to. Mirages of half written stories can and do exist on the metafictional plane of existence, but life traditionally begins when the story in some completed state meets its first audience. The audience gives their passion to the story, based on their understanding of the plot and characters, and that story then manifests on the metafictional plane of existence.
Even very shortly after being born, a character usually has self-awareness about their own fictionality. Most obviously, this is because going through a story, where one is a single person with a single set of experiences and one interpretation of events feels very different from living in a story's aftermath. The former is very similar to what a person in the real world would experience, the latter is essentially being comprised of many different versions of the same you, simultaneously. No matter how similar the interpretations, the difference is easily felt. Very rarely does anyone need to explain to a character that they're fictional. Like how a foal can run a few hours after its birth, fictional characters naturally begin feeling around the elements of their nature quickly.
Some characters burn bright and die fast. Your edgy DeviantArt Warrior Cats OC is likely not long for this world as you grow up and find other interests, but other stories will have a lifespan much closer to that of a person – or even longer. How exactly this manifests has a lot to do with the medium of their work, which traditionally only encompassed literature and performance arts. The oldest and most famous of these characters have portrayals built upon them layers of sediment, to the point where you can feel the myriad interpretations comprising them; so many famous actors and writers and storytellers coming together to create a varied tableau, a kaleidoscope of a preson. But even for more obscure stories characters experience these things: if they're non-visual their appearances shift around when not looked at, sometimes hard to notice even if it's obvious. If they're from non-auditory media they have strange voices that sound almost like multiple people talking at once, with different cadences and tones and pitches, but you can never pick out more than one person talking. These effects are known as polyphony and polysomny, respectively. Having an especially notable polyphonic voice or polysomnic appearance was regarded as a sign of health and vigour because it meant many people knew your tale and were imagining you. The reverse meant that you were either newborn or not long for this world.
Conceptions of aging have changed very rapidly as new technology has developed new mediums such as film and video games. What was once regarded as a sickly state of being is now recognized as somewhat normal, as media began to have access to video and audio to portray their stories. Characters appreciate certain art styles as cool or sexy without dwelling on the singularity of it all. It's not troubling to encounter someone with one clear voice instead of many. Ideas of what's healthy have been somewhat preserved by a communal sense of awe at the polyphony and polysomnia displayed by literary or franchise characters, but more out of a respect for age than the youthful health it originally signified.
There are two ways to be considered dead as a story, three if you count the combination of two causes. Theoretically, both can be recovered from, but it's much more likely in one case than the other, for reasons that are likely intuitively understandable. Stories subsist on the passion of the audience and a material for that passion to be derived from, and if either of those two disappear the story is in a bad state.
Traditionally, funerals for stories believed to be dying happen before death actually occurs. This is because when a story is forgotten completely by people in the real world, usually in cases of Deficit of Canon, fictional characters also become unable to remember most details about that story, almost everything fading to a blur. A story being told and a story being forgotten were deeply intertwined back in earlier days, so when people saw the signs of decay they moved quickly to make memorials while they still remembered the story in question. However, as media preservation has somewhat improved in recent years, it has become more common, albeit still rare, to have funerals after a character has been labelled societally dead.
Deficit of passion is fairly simple. Stories need to be read and heard and cared for and loved to exist; when people stop doing that, a story dies. Characters that nobody cares about anymore are reduced to a purgatory of thoughtlessly repeating their story it over and over again to a grand audience of nobody until the source material is either rediscovered or rendered irretrivable. They cannot interact with the rest of the metafictional plane consciously and do not remember it or their experiences, losing their self-awareness. Any attempt of conversation with them is essentially pointless, as they can't properly comprehend who or where they are. This is the more common death to recover from, since an old story being rediscovered and read again, prompting the new generation of fans is more common than finding a piece of lost media in its entirety, as in the other death.
And yes, much poetic has been and will be waxed about how death comes when nobody loves you anymore.
Deficit of canon is more complicated to define, because what constitutes a canon can be very vague compared to the standard set by modern day storytelling. Canon, in this setting, is essentially a shorthand for unified agreement on details relating to world, plot, or character. This sure could take the form of a book or television series, but oral storytelling has canon too because even though each telling of the tale is slightly different, there's still enough consistency to create a sense of cohesion overall. Stories can get by on shockingly little canon by modern standards – but even then, there's always a limit.
Most obviously this limit is hit when people know the concept or outline of a story, but can't find the original. The poster children for suffering from a deficit of canon are characters from lost media. These characters are in a sorry state: flickering in and out of being, they appear fuzzy or in shadow; it is supernaturally impossible to make out many identifying details. They seemingly don't comprehend much of what's going on around them, but it is still possible to hold a short, simple conversation, if you're lucky. They have no homeland. The only things tying them to the metaphysical plane are the memories in a few selected people and their passion to find any trace and piece the story together.