To replace your character model in the game, while playing:

• Enter command prompt mode by pressing the '~' key
• Type in 'DebugMode 1' and hit <Enter>
• Enter command prompt mode again by pressing the '~' key
• Type in 'SetAppearance #', where # is the number of the desired model (between 1 and 298), and confirm it by pressing <Enter>

Note: all commands are case sensitive.

RealmForge over at the BioWare forums was kind enough and patient enough to detail available models:

The keen-eyed amongst you may note that not all numbers are represented in this list. The missing numbers are generally humanoid models that can be created by customizing armor and clothing, without changing models.

If you discover any errors in this list or have information to add, please contact us.