In the earliest black-and-white Mirage Comics, Raphael was the most violent turtle and had a tendency for going berserk either in battle or when his temper flared up. He has a somewhat cynical and sarcastic sense of humor. In later issues, it's shown that he is not particularly fond of the supernatural, stating it up front in Tales of Raphael: Bad Moon Rising.
Raphael mellowed somewhat as the series went on, possibly a key moment for his character development was when he allowed Leonardo to go in alone to defeat The Shredder after nearly killing the shredder himself in issue #21 of Volume 1. Since then, he has been less likely to challenge Leonardo's leadership, and on the whole is more friendly towards his family and allies.
Of his three brothers, Raphael is actually closest to Michelangelo, even having stated it in the fourth issue of the original TMNT comics. Raphael often shows a caring, more laid-back, side of himself when around his youngest brother, frequently indulging the younger turtle emotionally when at home and ferociously protecting him from harm when in battle. Raphael openly admits that the mere thought of his youngest brother being grievously injured causes him to experience violent rage. In his self-titled one-shot micro series, Raphael met human vigilante Casey Jones, his foil, who was even more violent and unstable than he was. Despite their brutal first meeting, the two have since formed a close bond.
More on this character soon, for the meantime read his bio here: