In this video guide I will show you how to complete a stage only using a club, spear, or hammer. First acquire a few weapons of each in case the durability is low. This way if one breaks you have a back up. You can get multiple weapons by either buying them in the store, finding them in chests, or by taking them from enemies.
To take them from an enemy you have to have your fists equipped and the enemy nearly dead. Then you use the throw (RB) button to take the weapon from them.
Now I would recommend doing this on the first level on the difficulty Jack to make this a lot faster. Also note that there is no shop to repair weapons, that is why I recommended having at least a few weapons of each type. I would also save the weapon ultimate for the boss as well to end the level quickly.
An important note is that you can not kill any enemy with a throwing weapon or by using the throw move to finish someone off. Only use the desired weapon for the achievement. When killing the crocs on the rooftops use the jump attack, or deflect using the LB button.
This method can also work for the sword and fists as well. Once the level results screen appears your achievement will pop.