I've been a strong believer in that any part of information about a character can be relevant to growing them as a person. But I have a hard time deciding when to make what factors relevant and when to make them known. Lately I've been experimenting a bit, by showing rather than telling about a character. Having the characters think about certain things when it's relevant to the plot. How would I begin to know if a piece of a character's backstory/personality is relevant? Example: Cindy has a very specific hatred of dogs, but her hatred of dogs is (for the most part) irrelevant to the plot. But eventually the plot forces Cindy to deal with dogs and now she has to overcome it. I don't know when the 'right' time to bring up this kind of stuff in the narrative.
I was always told to try and forward the plot and character at the same time, rather than keeping scenes in that say forwarded character but did little else for the rest of the story. So, if you have a character that is a whimp and an up coming battle that moves the plot don't split those into two scenes. Don't have scene where we're told what a coward the character is, blend the two scenes and have him hide somewhere during the fight.
So let's take the 'hatred of dogs' example and look at a slightly different approach. She's stressed and on edge because of plot reasons and has an important plot decision to make and the neighbors dog just. keeps. barking. and god she hates dogs would they just shut up and let her think. Them being there makes the decision even harder and increases her anxiety (and thus the tension/level of conflict) without wandering away from the main plot. It's using the character trait as a spice to enhance what you already have, instead of an aside that moves the focus away from the plot. You can also choose to go the route of she is forced to deal with dogs and she must overcome her hatred, but be aware that by making it something she must learn to overcome, you're elevating it to almost subplot status, making it significantly more important, and will probably want to introduce her hatred of dogs earlier on so that it's not a surprise when it becomes a big deal (a bit of foreshadowing, as it were.) There's nothing wrong with adding in a subplot based on a character trait, but you won't be able to do so for everything you want to include. There also will be times when it's simply not necessary to include all the traits and backstory you have on a character. In a personal recent example, I had a character who was the middle child of 9 siblings and had drifted away from his family before the start of the story. I had a full-blown family history, all the siblings and parents' names, personalities, relationships with each other, and the protagonist, but...we never learn much about them in the story. We get bits of whistful thinking, snippets of memories when he fears he'll never get to go home and see them again, after near death experiences where he bitterly regrets not spending more time with them and appreciating what he had (which is relevant to his central character arc, and his central character arc affects the plot), but a lot of that family history just isn't important.
I don't think you should wait until she has to deal with dogs to let the reader know about her fear. It needs to be foreshadowed. Perhaps earlier in the story, she can fail at something because she's afraid to go near a dog. Then we can see a bit of growth when she overcomes her fear later on.
I'm having a hard time trying to understand this as anything but a contradiction. If Cindy needs to overcome something (be it an internal or external obstacle) at a point in the story, then that something must be considered relevant to the plot.
To explain the example, the fact that she hates dogs is kind of irrelevant for most of the story, so I'd have no idea when to bring it up in the narrative, much less HOW in some cases, if that makes sense? Prolly could have worded it better lol
Here's my opinion, if you want the scene where Cindy overcomes her fear of dogs to be impactful, it's better to have that fear shown or discussed earlier in the story as a sort of foreshadowing. Not foreshadowing it may not be a major issue (I've seen some people give some very negative reviews because certain things weren't foreshadowed at all) but I think the scene may not have the effect you want. Maybe you can hint at it earlier, and then during that scene (where she overcomes it) the fear is fully explained and then overcome? At least then it's something the readers would associate with the character. If it's not relevant to the plot, you could try and work it into a scene somehow but that's not easy to do. This is just off the top of my head: Cindy put down her calculus book and ran to the living room. "What's all this ruckus?" "Oh, hey, Angie's dog just had it's litter, do you want to adopt one?" "No thanks." [insert some action beat here] "Well do you want to pet them, they're so cute!" "N0. I-I've got to study," she said, and walked out of the room. Or maybe you could just have Cindy tell a story to some other character, but through a natural sounding conversation. Or maybe they play truth or dare: "Truth or Dare" "Truth" "What's your weirdest fear?" "Dogs" "You can't say that, everyone's been afraid of dogs since the Baxter's bought that Rottweiler" "No, I'm afraid of any dogs. Even being around Chihuahuas, sometimes even fake dogs, I just tense up" [include other characters laughing and giving Cindy a hard time] I just saw Die Hard the other night, because my wife never saw it, and there's a part where McClane is asking why Sgt. Powell isn't a beat copy anymore (over the radio), and he tells him the story about accidentally shooting a kid that had a toy gun, and how he didn't think he'd ever be able to point his gun at anyone again. Of course, at the very end, when Karl emerges from the burning building after everyone thought he was dead, Powell is the one that shoots him. They still could have left the part out about why Powell isn't a street cop and it still would be a great movie, but that scene would have lost some of the impact. It certainly wouldn't have called for the frontal camera shot of Powell holding the gun, at least not for the length of time it was shown.
Similar to the OP's thread about the Valkyrie, this involves a contortion away from character writing, because it makes the character passive to some extraneous detail outside them. The characters forward the plot. An it can't forward a character. An it can't forward a plot either - but there are plot devices which can help carry a character's choices into their external world. If you like, they are linkages and couplings that transfer drive from the engine of character. I should first discount the objection that overcoming a fear of dogs is character development. I think it's been exceedingly charitable of @Bruce Johnson and @Storysmith to discuss it as bona fide character development. Maybe it could stand for character development in a bad saturday morning cartoon - but because it makes the character passive to dogs, a fear of dogs is really a plot device. Giving Cindy a fear of dogs doesn't deepen her character, it might offer ways to cover over some situation in the story where (for example) her choice would be to go right up to Ted's front door and accuse him - but the author doesn't want to write that scene because it would resolve the conflict too soon. Plot devices are antithetical to character - they falsely reduce who characters are and railroad them into genre conventions. They might be necessary evils, but the worst ones are where the audience can see where the writer has dropped them in artificially, and how the writer then obviously relies on them to explain an incongruous character choice later on. Arguably, all of the writing outside the character's voice is plot devices - but the term is usually reserved for the convenient, and the artificial. If we notice ourselves inserting informational details into a character, it's a good discipline to try and edit them back out again and find more natural paths for the plot. I'd suggest that if the OP is wondering about when to insert a detail, there's still every opportunity not to. And to write the story without the fear of dogs. How would I begin to know if a piece of a character's backstory/personality is relevant? A well-written character's personality doesn't have pieces any more than a real person's does. I'll treat personality in the OP as another word for character, but if personality traits are treated as building blocks (e.g. "This character is brave and has a good sense of humour") it's unlikely to produce a relatable character - which is a common problem in anime and videogames. Real people aren't made by slapping a couple of adjectives together - and even cultural stock characters have more to them. A backstory places artificial limits on a character, so boils down to being a longer and more elaborate sort of plot device. At the positive extreme they are a way of establishing character motivation (Romeo was born a Montague), but at the negative extreme they can become a displacement activity taking the writer away from writing the character. One can imagine the bard posting on here:- I have this character. He's Italian. And he's in this noble house that's having a feud. And they have swords. He's good at climbing. Should I make him a bit gay with Mercutio?