Oftentimes, something will inspire me, such as a picture, stray thought, a story, or an idea, and a character will develop out of that. I'll let my mind play with the ideas for a while, putting the character in various situations to see how or if they would react (to determine what sort of world they would exist in, such as whether that squirrel works better in a medieval or a near future setting). Environment is a big factor in forming a character's personality, especially if the personality is something that's contrary to the setting (like the angsty warrior bear in Care-A-Lot or the happy-go-lucky sidekick in the middle of R'lyeh).
However, in the end, nothing actually beats putting pen to paper and writing down or drawing the character going through his or her paces.
For example, let's take my character
Deidrei Leafbright. She popped into my head when I saw
this picture in the
Ironclaw RPG core book. Her developing personality fit in with a group of female warriors I had developed before merging with an ill-formed gay character. Her character development really took off when I actually wrote about her adventures for National Novel Writers' Month a few years ago, where she developed into a plucky, intelligent, sneaky archer who found herself friends with a straight mage as she decided she needed to head out into the world to understand herself.
Sometimes it will take quite a bit of writing before you really begin to understand the character. When I got into the second novel, I understood more about her, revising the first novel to better fit who she revealed herself as. New revelations popped up, such as her pathological fear of ghosts or her need to help others stemmed from making up for being a childhood bully.
Putting characters into difficult situations for them will also reveal more about them. Deidrei likes girls and has trouble thinking of guys as anything more than mobile scenery. Then she meets the timid and shy albino squirrel Wally whom she takes a shine to. How will that play out, especially since he loves her? How about when I kill him near the end of the book?
In short, at best, a character will only appear well developed in your mind. By actually doing something with the character, by taking your impressions of the character and running with them, can you develop your character.