Bilious appears in the Discworld novel Hogfather, in which assassins are hired to inhume the Hogfather, the Discworld version of Father Christmas. On the Discworld, belief creates gods and other mythical beings (a philosophically fascinating idea that I haven't time to go into right now!) and without the Hogfather, the extra belief sloshing around starts to create several minor deities and folkloric beings, such as the Eater of Socks and the Jinglejingle Fairy. Among the beings created is Bilious.
Bilious is the natural opposite to Bibulous, the God of Wine. Bibulous is, of course, loosely modelled on Dionysus, what with his crown of vine leaves and constant partying, though he is also fat, which isn't especially Dionysian, and has its roots in modern imagery of excess. Bilious explains that the reason that Bibulous is so happy is that he never gets a hangover - they all come to Bilious, who is the Oh God of Hangovers - because when people witness him they clutch their head and say 'oh God...'
Bilious also wears a long white robe and crown of vine leaves - ensuring that he looks vaguely Greco-Roman, though the idea that Classical gods should wear long white robes probably has more to do with modern perceptions and children using sheets as togas than any actual knowledge of ancient costume. However, since I don't actually know much about ancient clothing either, I shouldn't pass judgement.
Bilious, as played by Rhodri Meilir in the Sky One adaptation of 'Hogfather'
Poor Bilious gets his revenge eventually - he takes a hangover cure put together by the wizards of Unseen University, and while it works to cure his hangover, the less pleasant side effects of the cure get transferred to Bibulous, who is interrupted mid-cocktail (he is enjoying some rather more modern luxury at the time, involving a cocktail with a slice of lemon in it and a rumba, and the timeless luxury of two gorgeous girls snuggling up to him). Bilious watches in fascination as all his misery is finally visited on the cause of it. It's very satisfying!
That's all I've really got time for now, but more thoughts on Discworld gods to follow at a later date. Also, I got the complete box-set of the BBC/HBO series Rome for Christmas, so once I've finished going through I, Claudius, I'll go backwards in time and start on Rome. Happy Christmas everyone!