The Greeks had no knowledge of how to perform amputation on living flesh. The big problem for the Greek doctors was loss of blood. Hippocrates does mention the use of ligatures elsewhere, but there's no mention of them used in amputation. Where he does mention ligatures he stresses the danger of gangrene. By the first century AD, a doctor called Celsus does write of both cauterizing wounds and ligature of veins.
The other likely cause of amputation was trauma. Guess what? Trauma is a Greek word. Again, the Greeks seem to have been remarkably conservative. It appears they'd rather have carried a mangled limb and risked infection than amputate. This might tell us something about the likely survival rate of amputees. Hippocrates mentions cautery, which means to apply burning heat, about ten times throughout his works, and a few of those are in reference to wounds, but never in reference to amputation. Nevertheless it seems likely to me that in the case of traumatic amputation, the wound would have been cauterized. Incredibly, they didn't use a tourniquet. The tourniquet wasn't invented until 1718!