DJ gets my vote for the most realistic character.
DJ's attitude is relatively believable given her sistuation (two younger sister, no mother, a dad who favours Michelle). As well, DJ's progression from 10 years old girl to 18 year old young woman is fairly believable. She always seems to act her age, not older or younger.
Becky is fairly believable, and would be my second choice for most believable character. The one thing I found not believable about Becky was that she was willing to move into Danny's attic with Jesse, and then stay there as long as they did.
Stephanie is portrayed as ultra-precocious, and while some kids are like this, Stephanie's precociousness is used as a means for her character to develop faster than is believable. There are hints of this in the middle seasons (Steph having Brett as a boyfriend; Steph's crush on Tommy Page), and it becomes much more pronounced in seasons 7 and 8. Though Stephanie is ages 11-13 in those seasons, she behaves more like ages 13-15. Not really believable.
Michelle is partially believable, as many little kids are like that. What is not believable is that she would be as well behaved as she was given Danny's apparent abrogation of certain aspect of parenting her.
Jesse is the most believable of the guys. His personality is more or less believable. However, his Elvis obsession, and his hair obsession, are both exagerated to the point of not being believable. Also, his career path is not entirely believable. It's not really believable that he would get to do the advertising job at home. Also, I don't find his ownership of the Smash club to be that believable, in that it's not likely he would get willed the Smash club in the first place.
Joey, Danny, and Kimmy are clearly over the top characterizations, and not intended to be believable. Joey with being a cartoon obsessed kidult; Danny being a obsessive-compulsive neat freak who thinks he's way cooler than he is; and Kimmy being an incredibly dumb girl with an overly garish fashion sense.