Almost all of the violent scenes are off screen or cut away before anything bad is seen. We sometimes see the aftermath of violent acts but nothing bad.
The opening fight of the games implies a few deaths, we briefly see blood splattered onto the ground and other objects, and throughout the games, we see open wounds (no longer bleeding) on many of the characters. A group of older teens travels as a pack and hunts down young children, there are a few just-offscreen stabs and kills. They find a teenage girl at night, alone, and we hear her screams, implying death.
A boy's neck is snapped, and we see a few people moments after death. An arrow sticking out of a boy's chest, a poisoned girl, her skin a light blue. One girl is killed by a flying spear and pulls the weapon out of her chest before dying, and another girl is swarmed by poison wasps and is killed by them. Katniss receives burns from a forest fire, a girl is slammed into a wall, snapping her neck, Peeta and another boy roughly punch each other, Katniss also receives a knife slash across her head. An explosion is triggered. One boy in the cornucopia bloodbath gets his neck slit by another tribute.
In the final battle, dog-like beasts attack Katniss, Peeta, and another boy. Katniss shoots one in the neck with an arrow to no avail. When the third boy falls into the pack, they begin eating him and tearing him apart (offscreen), and Katniss shoots him to kill him quickly, as a sort of "mercy killing"
Video footage shows a past Hunger Games, where a boy beat someone to death with a brick. We see the beating partially, with the blood covered brick being raised into the air and brought down hard, almost out of view.