Everyone tells stories about their lives, about who they are, about who they wish they were or who they think others wish they were.
Our characters should do it too.
Humans (or human behaving aliens/paranormal creatures/superheros ;)) tell stories about things that happened in their lives for pretty much two primary reasons.
1) To connect with others. If you share a bit about yourself, you are inviting the other person into your world, sort of like a window display. How much you share, and what, determines how close you and the other person are and at what stage in the relationship. Too much too soon could kill a romantic or even just friends relationship. Too little too late can make the other person feel you don't open up to them and again kill things.
If I want someone to think I'm adventurous, I might regale them with tales of my one skydiving jump, or that many of my ex-boyfriends were musicians (both of which sound far more adventure some than they were- ok, the jumping out of a plane all by myself was adventurous ;)).
If I want someone to think I'm smart, I might talk about my time in Mensa- ok, that would be a lie, but you get the point.
Most story sharing isn't that pointed though, people aren't always calculating what they want the other person to think. Most of the time something in a conversation triggers an emotional response and off we go with a story.
2) However, the second reason is more controlling. Sometimes either insecurity or ego step in and people create, rehearse in their heads, and repeat stories which reinforce a persona they wish to project. These can be complete fabrications or simply an embellishment of what really happened. "Did I mention the 30 times I've jumped from a plane? Lemme tell you about time 12 where....." I've jumped once, but if being adventurous was part of the persona I wanted to project, and I felt insecure around this person--that one time may have jumped-LOL-to 30.
Braggarts will often do this as well, and oft times for the same reason- insecurity. Eventually, the made up and rehearsed stories become real to the teller. They've told it so many times it is real- to them (Gollum and his "birthday present" anyone? ;)). What would happen to someone with a rehearsed set of lies, both internal and external if a new person in their life burst those lies?
The same with our characters. What does your character share with others? What don't they share? You could show alot about how a character feels around different people by the changes in the story. Do they lie to everyone simply because they feel inferior? Are they trying to cover up a failing? Do they fear their friends won't like them if they know the real them?
All of those things and more can be shown by designing the stories they tell.