My main protagonist has a complicated history and relationship with her family and I'm having trouble deciding how to reveal this in the story. You can't just dump it on the reader in one paragraph but it is not exactly a mystery that should be solved slowly either. In a lot of stories, the main characters have a normal life so not much is to be said about it. The characters with gritty backgrounds are characters that the MC interacts with. I don't want to start at the beginning of my character's story, since it will slow down the plot and is unnecessary. It shows how she first meets the group she is with but it is unimportant how they met. So how do you best reveal aspects of the MC's history? And how do you best explain the relationships they have with the other characters?
By having them interact with other characters in tenuous situations. It's an experiment - put a chemical into a testing room and heat it until it cracks, or refuses to respond. Through dialogue and interaction with other characters, people will be able to make assumptions about your character's status. Or take the perspective of one of the characters and let the reader delve into said character's thought process.
I do something similar to the above. Usually I just bring up bits and pieces of their history on a need to know basis. When they're in a situation where it needs to be known I show it either in their interactions or via brief flashbacks.
Ask yourself if the backstory is essential to your story. If you find it is, you're right, you shouldn't create huge dumps of information. Anything that you deem unimportant (such as how they met), cut. Reveal backstory as you go along and only as needed. Don't 'explain' the relationships between people, make them obvious by the way they communicate with each other etc. Readers are not idiots, well not always.
Read the way that other people do it; as has been said above, reveal the information on a need-to-know basis. The way that I generally advise is to only do it when the character is thinking about it. The problem with flashbacks is that it assumes that the character is thinking about the entire thing with a certain level of detail. That's not how it works. L. Lee Lowe (author of Corvus (which I read a while ago) and Mortal Ghost (which I'm reading now)) is spectacular at this sort of thing. You get pieces of a puzzle, and eventually, you learn to put it all together as the character reveals it all. So it's not some huge dump of information. It's real.
As others have said, don't dump information. Add hints from various interactions through dialogue and actual physical action. Sometimes characters' habits can also reveal a lot about them. For instance, let's have a stupid example and say the character in question always blinks quickly when he sees a rat. Well, maybe when he was little rats ate his parents or ran down his house or kept stealing his homework or something. But that can easily be first hinted with the blinking, and then later revealed (though you can't, of course, say outright, "Oh yeah, Mary, did you know that I blink a lot because it's a natural nervous reaction I have to mice since I'm scared of them?).
Don't. Write story, not back story. Reveal what is absolutely essential about the character's history as it relates to your story. There will be a time in the story when that nugget needs to come out. If that moment never comes, you don't need it.
