When and how did life on earth become carnivorous ?

The very first organisms must have dined on the organic chemicals in the primordial soup. As these became scarce, organisms began extracting energy from inorganic chemicals. These bacteria are still around, as they feed on and derive energy from sulfur and other inorganic sources. Only when bacteria became abundant can other organisms make a living eating them.

I presume you are asking about multicellular organisms. The first appearance of multicellular animals can be traced back to a short time before the beginning of the Cambrian, which began 540 million years ago. Before the Cambrian the only animals that might have existed were probably jellyfishes, as there were disc shaped fossils that suggest they were jellyfishes preserved in the Precambrian deposits. The jellyfishes might have fed upon other simple aniamals like worms. After the Cambrian radiation, the number and variety of multicellular organisms increased. By the time the Burgess Shale fauna was preserved, during the mid-Cambrian, a bewildering variety of carnivores and predators had evovled. Anomalocaris and Opabinia are two of the well known Burgess Shale predators that have been described. Therefore the first carnivores were likely jellyfishes, followed soon by other predators which evolved during or shortly after the Cambrian explosion.

Well I can't pinpoint an exact time but I would say in the early stages of life on Earth is a good estimate. There are single cell organisms that eat other single cell organisms on Earth now, so we can postulate that they existed then as well.

EDIT:

It is also pretty safe to assume that the single celled organisms that could eat other one were pretty successful, and that is why they still exist now as single celled organisms and as multi-cellular organisms. Herbivores spend their entire day foraging for food since plant material is hard to digest and is relatively low in convertible energy. Carnivores can spend much larger portions of their day doing other activities since meat is packed with convertible energy. This presents a distinct advantage for both. Herbivores always have a readily available food supply, if the plants start dying in an area you just move to the next. Carnivores have to work for their food, and follow the herbivore population; they eat less often but when they do they don't need to eat as much.