There are no mechanics as such, but every White Wolf game has a section in character creation about breathing life into the character. This is just a list of questions that help players think about their relationship to the setting.
They also encourage you to play a short freeform session with each character individually. This prelude usually includes the time that the character changes from being a normal everyday person into someone special - the Embrace, the Change, the Awakening, the Second Breath and so on.
In your situation, starting with a few questions from a free-form roleplay questionnaire is a wise starting point. Edit it so that it has interesting questions (how is eye color interesting for every character for instance?) as well as questions that directly relate to the themes and mood of the games you like to play.
For instance, if you're playing cops with integrity, you'll want to ask something like "When have you ever been tempted to ignore Miranda rights and what happened?". For a game about financial moguls on Wall Street, "What was your first deal and how did it make you feel?" is more appropriate.
Focus on getting qualitative answers about why they made the decisions they did and what they felt about it. These help provide more insight into the characters as they go along.
It's also OK if they don't answer all the questions. Answering one or two will give enough juice to get started - and they can always fill in others later.
System Specific Mechanics
There are many recent systems which do create backstory in characters. Two examples are FATE and Apocalypse World and their descendant games.
FATE powers Spirit of the Century and Dresden Files as well as a CORE version. As part of character creation, each player names and details a story they have already completed and another player is chosen to guest star in it. The rules from Spirit of the Century might help you create a process you can use in your game.
Apocalypse World (used in Dungeon World and Monsterhearts) provides a history round to help build a world between the group. Each archetype has a number of items that can be used to tie you to another character.
These approaches help build an existing group cohesion by defining characters in terms of each other.
As the simplest thing you could do, go around the table and ask each player for a statement of a positive relationship with another character and a statement of a negative relationship with another character. Each statement should give some idea of how it came about and why, helping add details to the game. "X saved my life in an ambush in Iraq." and "Y snores like a woodchipping plant, I hate sharing a room with him but won't tell him why."
Be careful with the negative relationship though. This should be on the level of things you don't like about friends, but accept anyway. If it gets to "Z stole my wife and I hate him for it" then the game is going to be about intra-party conflict. That might be ok, that might not... but steer clear unless you know your group can handle and enjoy it.