It generally depends as to where the bio is bein' typed, whether it be in-game or here on the forums/some other website.
In-game, Vespian tends to start off the biography with my character's name (full name, preferably). Then underneath that, Vespian will put that particular character's chosen subtitle (e.g, for Vespian, it's The Mad-eyed Devil). The subtitle serves for role-play purposes mainly and to differentiate my characters from others who are addressed by the same callin'.
Underneath that, I will lay out the basic details that I deem important to know (e.g, gender, age, species, if they're betrothed to anyone, relatives that are alive, any possible descendants, etc.) then, following suit will be any acquaintances that character has made throughout the time I've spent role-playin' with them.
With some characters, such as Yusei, Vespian will add any physical disabilities the character may be endurin' but normally such details will only be laid out in the biography if it's somethin' more severe than just an inflicted wound. Some examples would be if the character suffers with anemia, has occasional spurts of amnesia, if they have a weak constitution or eating disorder, etc.
Then comes the fun part of addin' a selective few "Fun Facts" as I would like to call 'em regardin' anythin' I deem necessary for the reader to know. These tidbits are normally one sentence per fact and they are very short and to the point of what I'd like to get across. The content they carry usually range from how the character may act, somethin' they like/dislike, any phobias they may have, their favorite hobbies, etc.
Scroll down a bit and you'll come across links that will direct ye to the character's preset (since all of my characters have presets), the character's natural appearance drawn by a lovely artist (to which I will credit; this is for those characters whose appearance may differ a little than what the FeralHeart models provide), a link that will direct them to their full biography (since ye're only able to fit so much about a character in the in-game bios), and sometimes even a link that will direct them to their species' information (for characters who are of a species originally made by someone).
Lastly, I'll add the copyright sign and year as to when I crafted the character to indicate that the character and their biography are mine and mine alone (e.g, Vespian ((c)) Dishonored; 2013).
If I have room after typin' all of that in, I may go back and add a fittin' quote that the character may say to reflect who they are to onlookers (e.g, for Vespian, his biography quote is, "I am the beast that stares into Heaven with a thousand eyes.")
As for biographies written on the forums/any other website, well they tend to be the same only much, much more detailed and decorative. I like to add photos of my characters drawn by other artists here as well as a legitimate, detailed back story that I do not bother much to try to summarize and squeeze in on the in-game bios. I'll also add more basic information such as what their reputation may be, what organization they were a part of last, a theme song, etc.
Along with that, I'll add more fun facts and character ties that I may have been unable to fit within the in-game bio.
As stated, there's little of a difference between the in-game and forum/other website bios other than one is ten times more descriptive and actually provides a typed out back story containin' a character's origin and history while the other is more blunt and not as wordy for those who would only like to know the present details.
