Samples of applications we particularly like can be found at: Sample Applications
A few thoughts to help you make your character:Remember that in comparison to the modern day, the UFP is a utopia. Though there are characters from bad worlds (see: Tasha Yar) these are typically far-flung colonies. The average Federation citizen has led a good life, with plenty of food and a far better education than the average modern-day person even in first world countries.
Starfleet Academy only accepts the best of the best. One version of their test takes four applicants, of which only one will make it through, and even then none may measure up. True, applicants may re-try, but Starfleet's standards are exacting. Only the best will get in, and only the best will make it to graduation.
Starfleet's mission is to "explore strange new worlds" and etc. Most Starfleet officers are dedicated to this, even after the Dominion War (perhaps moreso because of it!) Learning, discovery, and exploration are prized as high values in the Federation and Starfleet. Even a Security officer will likely be in it for the thrill of discovery and be reluctant to use their phaser unless neccesary.
That said... it's your character! Make someone interesting, someone compelling. Make a character with goals, motivations. Maybe your character is like Tasha Yar and has had a hard childhood, or is the first Ferengi in Starfleet. We don't want to read 5 paragraphs on the details of your character's childhood--those things will come out in play, where appropriate. We *do* want to know who this person is, where they've been (in general), and where they're wanting to go.
Think of making a character for a Star Trek show, a protagonist. This is not your ordinary person. This is someone extraordinary, someone worth the time the camera will spend on them. Don't try to wow us with how much you say, though--just say it well.