I am about halfway through my story, and I realized I haven't brought in very much of my character's backstory. Any tips on where to bring that in, and how?
There are several ways to bring in backstory.
In the first Harry Potter book, the first chapter occurs ten years before the second chapter; Harry Potter is the boy who survived; somehow. A magical tone is introduced, the orphaned baby needs to be put with his new foster parents. It is the first time we understand that Harry Potter is not just another kid growing up to go to magic school, he is unique above all the other kids.
Try to not tell the backstory. Show it if you can. Readers can accept that characters have wondrous skills. In Butch Cassidy and the Sundance Kid, Sundance is an incredible sharpshooter with a pistol. He's trying to get hired as trail guard, the trail master wants to test his shooting skills: But insists Sundance point, take aim, and shoot at a tin. Sundance misses, then says, "Can I give this a try my way?"
The boss says "Sure," and Sundance holsters his gun, then like a gunfight draws and shoots the can, sends it flying, and shoots it again in mid-flight, and then again. The boss nods, "You're hired."
No explanation of where or how Sundance learned to shoot, ever.
A lot of writers do a lot of background work to justify their characters, and then feel compelled to share this creativity with the reader. But you don't have to do that. The work is worthwhile, to build a consistent character both emotionally, and with specific skills and limitations so you can avoid just adding new random skills to get them out of a tough spot (deus ex machina like).
But in Ocean's Eleven, where Ocean assembles 11 master thieves for a fantastic heist, we are barely aware of backstories. In one case, I think the pickpocket is described as the son of the best pickpocket Ocean ever knew. That's all the backstory we get on that guy. The cardsharp, we just see him at work. The acrobat, we just see him at work.
My first advice is, if you don't have a plot need to talk about backstory, just don't do it.
If you can, get away with a throwaway line, and leave it at that.
"Where did you learn to shoot like that?"
"Far as I know, I was born with a rifle in my hands. Not much else to do in them hills, but shoot and reload."
If it is not critical to the plot, leave it out. In Mission Impossible, we don't know the backstory of Ethan Hunt (Tom Cruise); how he grew up, what influenced him, how he became the Impossible Mission guy. He just is.
In an episode where we do need to know some element of his backstory, make it a turning point in his life and show it in flashback. You can just have a flashback chapter, subtitled "Twelve Years Ago." And remember, chapters have no minimum length. Always try to put such flashback chapters in a low spot in the plot, the best is places where the characters have some down time anyway. Places where you might have just skipped to the next day or week anyway.
It is always better to show backstory (in a scene the reader can imagine and vicariously experience) than to tell (in dialogue the reader should just remember). If you must, ensure there is a good reason for somebody to ask, and then try to keep it brief.
In many instances, we can take backstory for granted. I loved Mozart's music and skill from the first time I heard it, and for a decade before I learned his backstory. I admire actors, and magicians, and musicians and singers and athletes and writers, without knowing a thing about their backstory.
Your backstory is good to keep your characters consistent and coherent (their personality traits "fit together" as a person), limited (so you don't stray into deus ex machina territory), and on track with well-defined motivators.
But backstory is like underwear; good to have, but you don't have to show it to everyone.
First you should consider if the backstory even needs to be included. If your story so far hasn't suffered by leaving out the character's background then it might not be necessary for the story you're trying to tell.
If you decide you really do need to include it, then probably the most widely used narrative device for inserting backstory is the flashback. The narrative moves to an earlier point in time, and often to another place as well, frequently telegraphed by framing devices such as characters reminiscing on the past or reading a journal, listening to a story, and so on. The flashback is written as part of the story, rather than a summary, so the reader is transported to that time and place and shown events directly.
The risk with flashbacks is they can have a "derailing" effect on the main story if they go on too long, or you have too many of them, or they wind up being more interesting than the rest of the story. But if you find your flashbacks are the most compelling part of the story, that might be a sign you need to rethink what your story is about.
Monologues, Soliloquys, and Summaries
So if the flashback is the "showing" option, then the "telling" option would be a summary. The summary can take different forms depending on the narrative perspective.
A third person story can make use of its (limited or unlimited) omniscient narrator to dive into a character's past for an objective (or at least outside) perspective on what happened. A first person account can make use of soliloquy as a form of summary, where the first person protagonist narrates their backstory to the audience, making full use of their distinctive voice to color the telling in interesting ways a more objective third person account wouldn't have. Then you have the monologue, which can be used in either first person or third person stories, where the character actually tells their story to another character.
Having the character narrate their own backstory through monologue is a great way to deliver this kind of exposition if you're writing in third person, since it allows you to make use of the advantages of first person narration while maintaining the third person omniscience. You can allow the character to embellish or downplay parts of the story, inject personal feelings, and contrast this with more objective information obtained through narrative omniscience.
Probably the biggest advantage in this is the existence of other characters listening to the story alongside the audience. Even if only one other character is present, they can provide feedback to the story as they hear it. Comments and reactions to the story by the characters listening to it help make it feel less like an exposition dump and more as an experience your characters are sharing. With an omniscient narrator you can peek inside the heads of those other characters and get their internal reactions to the story as well.
It's up to you how you want to do it, but a monologue gives you a lot of options for developing character relationships and personalities. A soliloquy is a good way to do it for a first person story if you want to keep it brief, with a third person summary also being on the table if you just want to get it out there and over with. And if you really want to dive into it for a full narrative experience, consider doing a flashback.