Burking is to kill by suffocation or smothering(this leaves no marks on victim's body)to sell corpse. Like erajshazad mentioned this term is derived from the two murderers William Burke and William Hare.
this is a new term for me too. i read somewhere that Burking is a homicidal smothering technique involving violent asphyxia. The term 'Burking' comes from the renowned criminal 'Burke,' who, with his accomplice 'Hare,' murdered elderly individuals using a combination of smothering and traumatic asphyxia and sold the bodies to Edinburgh's medical school.