In order to kill the carbon chain, you'd have to kill all the plankton, which would destroy the oceanic food chain pretty quickly. Plankton are the primary produces of the ocean, much as plants are the primary producers on land. Kill the primary producers, it's not long before everything else dies. I'd put it at a handful of years, somewhere between 2 and 5, with the carrion-eaters being the last to go.
You can't have a solely carnivorous food chain thanks to conservation of energy. The energy input into our food system comes from the sun, whose energy is captured by plants and plankton. Without that input, you have a closed system which will rapidly dwindle as it consumes all its energy. Large predators, for instance, have massive caloric requirements which would quickly deplete (now-foodless) prey and starve out the species as everything they eat would either starve to death or be eaten.