I want to tell a character's backstory, but I don't want that character to tell it directly to the protagonist, or to use another character to do it for them. Is there a way to do this?
In addition to the advice of weaving it in, in pieces, I'd suggest trying to add it not through exposition, not even exposition in dialog, but rather through action and then an offhand not-quite-exposing remark.
Example: Backstory - Trixie had been trained as a ninja, a secret weapon for the other side, she was their ultimate assassin.
But we don't know that. We just know that:
Trixie is a seemingly sweet twenty-something. She's tagging along with Pancho for reasons we don't know. He's on a mission to save the resistance. He's the guy with all the amazing power. They get in a scrap with a bad guy, who throws a knife at Pancho, but Trixie catches the knife mid-air. (Bad guy runs off.)
Pancho says, "Whoa. How did you do that?"
"You'll find I'm full of surprises," Trixie replied.
^ No exposition, except the amazing ability to catch the knife mid-air (which is shown.). But now Pancho wants to know (that's some tension.) Of course the reader does too.
Pancho starts to nag her, but she won't tell him (tension).
Eventually, other hints come out, through a variety of incidents, and there's some sort of nice reveal chapters later. Dribbling out the backstory gives the reader the sense of progress in addition to the other plot points that are progressing.
This backstory should be integral to the plot, but I assume you have that covered.
You do it in chunks. Let the character explain that he is the right one for the job because he has done this thing in the past. Let another character a chapter later point out that they knew this character for years and can always trust that this character will get it done. Let him share something at the fireplace from his past in exchange for information about the other character. Another chapter later you see this character do something incredibly good - another character murmurs that it must have taken him years of hard training to achieve this mastery.
You don't want to simply tell the reader in a long monologue, but you need to show it somehow. As long as you don't dump the information on the reader and instead introduce it whenever it's relevant this is fine - it doesn't matter whether the character explains it themselves or another character or someone just thinks they know something. And you should only ever show your reader the important parts of the backstory. It doesn't matter how many sisters the character had - except for when finding one of them is his motivation, or they collectively taught him something, or they are individually important to him or the story.
I love the answers already given. But I find something missing: hints.
There's no rule that says you have to tell or show much of anything about someone's backstory. You have to know it, and you had better know it. But you don't give 4, you give 2+2 and let your audience figure it out.
She stood there, eyes angry as she clutched her right forearm, the scars were somewhat faded but all too evident under her touch.
This makes the reader want to know. Why is she angry, what happened to her arm, what injured her enough to leave scars.
Think of it like breadcrumbs in the forest. Some readers will ignore them and admire the trees, the chirping birds, the rustling leaves. Some will obsess over the breadcrumbs, it will tease their curiosity and make them crave the answers to the little puzzle pieces you give them over the course of your story.
Every time Cassie looked at her, her arm itched. As bad, sometimes worse, than her recovery.
Now we have a name to work with. Cassie. How is she connected to the injury? What is their relationship now, what was it then?
This way, if you lay your crumbs right, with little windows into the past, you can show the backstory of little things that forged your Point of View Character, or whoever this is about, into the person they are now.
A great example of this can be found in how JK Rowling wrote the Harry Potter series. She leaves the breadcrumbs for any who is looking to find them, but she nestles them between so many other things going on that you sometimes only find them after reading the book again and again.
The sole lightbulb in the room flickered. Off and on, on and off. Suddenly, she was no longer in the dank and gloomy attic.
Cassie stood over her, baseball bat in hand, hoisted high over her head with a crazed look in her eyes.
Her chest tightened, tears stung her eyes as she held up her right arm.
Gasping for air, she jerked back from the ghost that lingered. Sweat poured down her pale brow, eyes desperate to find herself alone.
And now you know. The tale of her injury. The tale of the scars in her mind, and not just her arm.
Simply think about how you learn the background of other people in real life.
You get some hints from what people do, what others say about them, and from the context in which they appear. Then you get to know them, but at first your don't directly ask them for the story of their lives (unless you're a clod). Nevertheless you learn more about them from talking to them, and eventually they will tell you more about themselves, if you get closer.
Fictional characters should be described the same way. If you happen on them the first time, the narrator shouldn't break the narrative to draw up the character file and give it to the reader. Instead, the reader should learn about the character as this person interacts with their environment.
Getting to know a character, should be a process that runs along parallel to the storyline.
Mostly, you'd want the other characters to provide the backstory, in bits and pieces, relevant to the main story. But in a pinch, have the narrator (or point of view character) fill in what is necessary.
In this case, you would be "explicitly" informing the readers, but not the protagonist. The protagonist probably doesn't need to know the backstory (until the end) but the readers may. Sometimes it's best to have the readers slightly more informed than the characters (although you'll want to let them discover "most" of it for themselves).
This is another version of this question: Intentionally leaving out a part of the story, for a more interesting reveal?
In other words, it is a how do I tell something given that I have chosen a narrative point of view that is not suitable to telling it.
The answer is, you don't.
In the default narrative mode of western literature, which is third person omniscient (in other words, the author says, I am the narrator, telling you this story), there is no limit on what the author can tell the reader or how they can tell them and the only responsibility the writers has it to make the telling interesting.
As soon as you choose any other narrative mode, you are placing limits on what you can say without doing something obviously false. That is fine as long as you accept those limits. But if you can't live within those limits, then the only non-false way to escape them is to switch to a narrative mode in which those limits do not apply.
The most limited narrative mode of all is first person stream of consciousness. (In other words, the narrator is pretending that they are not narrating a story at all, but merely reporting their experiences as they occur.) The only reason to adopt this narrative mode is because your story is served by staying within the limits it imposes. If you have chosen this mode, you can't fill in backstory by any mechanism that will not obviously feel false, will not obviously involve behavior or a chain of thought that the narrator would not naturally engage in at that point of the story.
So if you have to get that information in to make the story make sense, choose a different narrative mode.