"Bollocks" is, indeed, a British slang word for testicles. "Billy Big Bollocks" is a slang term for any man who is acting braver and stronger than he actually is. He has a big mouth and is pretending to be a "hard" man. He is picking a fight which he has no chance of winning. I think Americans call such a man "Billy Badass". What book are you reading, Katrina????

Bollocks are testicles. So basically this means (if it wasn't meant to be literal): check out how cool\brave\manly etc Billy is. But here, since all the words are capitalized, it must be his nickname. Consequently he is considered to be a tough guy in his circle.