To find the story of Beyond Birthday - whose story was only mentioned as 'the LABB murder case' in volume 2 of the manga, and his death being mentioned in the Death Note timeline found in volume 13 as 21st January 2004 (2007 in the anime storyline) - you should merely read the light novel by NisiOisiN, called "Death Note: Another Note", in which all of the details of the LABB murder case can be found, including all existing details concerning Beyond Birthday's backstory.