I think the worst way to approach this is to play a caricature. For example, the blatantly oversexualised bisexual female. Often, that's what young male role players go for, and it inevitably turns out as a farce. Unless you can pull it off, I wouldn't go for the flamboyant fabulous gay male either (if you can pull it off, it might be excellent - in a small familiar group of good role players). Dorian in Dragon Age III is a decently written gay male protagonist, on the fabulous side.
One thing I would take into account is how the society views LGBT people - the degree of acceptance, legislation, etc. In one society they may be persecuted, while in another, there may for example be a highly regarded priesthood of a gay god, of course accessible only to homosexual men.
A character from the former society may be very private about their sexuality - or overcompensate and become openly celebrant about it when away from home. Meanwhile, a character from a sexually open society may be nonchalant about their sexuality and think nothing of it.
Regardless of orientation, characters will seek the same things from their object of desire - love, sex, companionship, understanding, stability and support, family, entertainment, friendship - feel free to continue the list.
On a side note, sci-fi and fantasy offer an additional avenue to explore: characters can have non-standard genders due to magic or technology, or due to being of a non-human species, and may change from one gender to the other, with full biological functionality. This may happen willingly or unwillingly, providing for abundant plot devices.
But, as Scras said, all these are only meaningful if they are going to be explored, as plot devices and character development.
