Do you have a friend, family member, or significant other that you don't understand fully or don't know the little things about? A lot of us do.

[edit] Steps

Make sure both you and the person/people you are trying to get to know are open enough to tell you things.
Think of things you would want to tell about yourself (for example: "I still suck my thumb" or "I like to mix my vegetables with chocolate." Whatever small things people don't know about you.)
Don't allow a lot of discussion during the activity. It can be hard to keep sharing if you hear a comment on everything you say.
Decide on the rules for your activity. For example, do you want a time limit. Do you want to allow anything negative/positive or just stick to neutral tidbits? Do you want to have a certain set number of times each person is allowed to discuss a fact? If so, how many and how restrictive do you want the discussion to be? (Do you want them to be allowed to discuss anything other than that specific fact during that time?)
Write your rules and guidelines down somewhere you can display them for the participants.

[edit] Tips

Allow people to request topics for facts but keep in mind that you can refuse to discuss it.
Create a list of possible topics to break the ice and post them with the rules.
Always start the activity yourself. You decided to do this, and the others may be self-conscious or unsure so you should get them warmed up.
Start with something surprising but not too off-the-wall. You want them to get the point that this is to share but don't make them feel they need spectacular things to say.

Don't tell anything major or embarrassing if these aren't people that are good at keeping secrets or that you're not sure if you can trust
Don't let it turn into a contest...or don't invite people who are going to get competitive. It's amazing how something like this can turn into a competition to see who has the weirdest things to say or something.