Definition of characters beyond a role of 'talking wallpaper' and giving them some depth is an-often opined wish of writers and game creators alike. Here's a perspective based on my own experiences and drawing from varied sources. The time you give this process varies according to how involved the character is going to be in the story. The more you involve them, the more time you can spend on them.
What are they doing?
Corollary to that: If they aren't doing anything why are they here?
What do you want them to do to your audience? This can be on an emotional or intellectual level; conveying a mood or theme.
Corollary to that: If they don't, what needs to change?
What capabilities do they have? This defines how your creation affects the world on their own (yes, a large army of flunkies is an attribute - and a character in their own right) This may be a literal ability to break heads or a more subtle persuasion. Your purpose and impact will suggest attributes but not everyone is the ideal tool for the job or an embodiment of betrayal; if you can build an element of conflict, this will give the character depth and determine certain actions.
What do they want? As with attributes, your purpose and impact may suggest ideas and again, conflict can bring depth as long as the attributes and actions of the character reflect the impact of the character. Consider how intense the motive is and what attributes the character has to get there. If you give the desperate father a knife, is he going to turn on the assassin or is he going to bide his time so he can cut everyone's bonds so his family can all escape?
How do they feel? Is your relentless killer driven by hate or by fear? Again purpose and impact suggest obvious courses. Attributes and motives may suggest common emotional states for the character - if all your noir detective wants is a decent night's sleep, how do they react to another murder and the police knocking on his door at 3am? Conflict with motives and attributes can be powerful if done in moderation - have you ever wanted to do something scary?
6. Inferences & Relationships
Now you have this raw data, take a closer look at the last three. How did they get them and can you make possible connections? If your knight rides, does he ride socially and care for his horse or is that a squire's role and beneath them? Is the relentless killer driven by fear because of his physical weakness and social inadequacy? This provides backstory, versimilitude and may even suggest events to drive the purpose and impact of the character.
7. Review for fit
You've got a character with history, motives, emotions and the ability to affect the world - do they meet your purpose and intent for them? Then look at motives. This sense-check can throw inconsistencies to light - why would that glamourous socialite visit a grungy bar? Is she on a dare? Looking for some big lunk to protect her? A check for consistency can do wonders in either suggesting storylines or making sure you don't create a turkey.
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