Determine which characters will be the most important ones in your story -- your protagonist, your antagonist, and any important supporting characters such as a love interest or mentor. These are the characters whom you will need to flesh out more than any others and whose decisions will drive the story forward.
Brainstorm a list of the traits that each character will need for your story idea and the character's role in the story to work. For instance, if you know you want your story to revolve around a son avenging his father's death, you will need your protagonist to be at least somewhat vindictive (to seek revenge), ruthless (to do what it takes in order to obtain that revenge) and imaginative (to come up with a revenge plan that your readers haven't seen before). If you want a character to serve as a romantic interest, on the other hand, she will need to have traits that complement the protagonist's and make her an attractive partner.
Determine your characters' goals. This is particularly important for your protagonist and antagonist; your protagonist's attempts to reach her goal will drive the story forward, while the antagonist's attempts to reach his opposing goal will create conflict and tension.
Brainstorm possible motivations for your characters. Why does your protagonist want his goal so badly? Why does your antagonist want to stop him? Don't settle for easy answers; "because she's a greedy monster" may cut it for a purely action-driven story, but it isn't a strong enough motivation for a character-driven story. The antagonist may be motivated by greed, but you need to know what she means to accomplish with all the wealth she's accumulating, whether it's a tangible goal or simply a desire to never again be as poor as she was in her childhood.
Write a history for each of your characters. What happened in their pasts to make them feel so strongly about their goals? A character questionnaire or survey may help you to answer questions about your character's life.
Determine how you want your protagonist to change by the end of the story. For a story to be character-driven, the protagonist must develop as the story continues, and the events of the story must serve as a catalyst for the protagonist's self-discovery. What do you think will happen in the story, and what flaws or insecurities in the protagonist will this address?
Write down your characters' goals and motivations, as well as how you wish your protagonist to change, and tape them to your computer screen or writing desk. Refer back to them constantly to make sure you are not having your characters act in uncharacteristic ways and to ensure the story revolves around your protagonist's personal change and self-discovery.
Give your characters room to breathe. For your story to feel character-driven, your readers need to know your characters' goals and motivations, which means your characters need to have time to reflect on them. If you hurtle frantically from one plot point to the next, your characters will not have time to think or talk about their goals. No matter how much effort you put into fleshing out your characters, it is wasted if you cannot convey their traits, goals and motivations to the reader.