In order to get the punched out badge in bloody battle, you must first complete the following tasks:
1) Kill 100 zombies in a single game. This can be done in any mode, but will likely be easiest in solo play.
2) Use only melee weapons to kill zombies for an entire game. Again, this can be done in any mode but may be easier in solo.
3) Completely deplete your health bar by taking damage from zombies. Once your health is empty, you will need to use a first aid kit or similar item to fully heal yourself.
If you complete all three of these tasks, the punched out badge will be yours!