Sometimes you don't really need to develop a fully fleshed character through a story, but rather a narrative to set them up to continue through in-game roleplay. If your character is joining LSPD you're not really as confined as you would be say if you wanted your character to be a member of an LCN organisation. So you've got a lot of flexibility in terms of where you can go with it.
Most important areas imo are their history - where they grew up, family, major life experiences, and then you might want to think about an arch/point of note where the story may pivot. Throughout the s