There are a few distinct methods which require few or no tools.
1) Skewers : good for fish and small pieces of meat, especially more tender cuts which don;t require long cooking. simply thread the meat onto a sharpened green stick and support over the fire. Here a fire of hot embers with little or no flame works best. The larger scale version of this is spit roasting.
2) Cauldron : if you need to boil water or make soup or stew but don't have a metal cooking pot there are two techniques. Firstly A shallow bag made from leather or tightly woven canvas can, with care be used to boil water over a fire without burning as the water seeping through the material keeps it cool. The second method is to use any container which will hold water, even a natural hollow in a rock, fill it with water and place into it rocks heated in a fire.
3) Pit roasting/BBQ : this technique is still fairly widely used. You basically dig a pit, line it with rocks, light a fire, let it burn down to embers. Place your wrapped meat in the hole and bury. This can be an efficient way to cook large joints of meat.
4) Wrapping/steaming : wrap the meat well in leaves, seaweed and or/clay and place on hot embers, this will steam the food and works well with fish. Directly wrapping in clay can also peel off feathers, scales or spines and so save on some preparation time (results may vary) .
5) Ovens : if you have a bit more time you could make a simple oven from clay or stones. This is essentially a hollow which you light a fire inside alloy it to burn down and then rake or sweep out the ashes and place your food inside.
6) Hot rocks : you can fry food which only needs quick cooking such simply by laying it on reasonably flat smooth rocks heated in a fire.
7) Direct cooking : some things like shellfish, thick skinned roots and tubers and even some cuts of meat can be cooked by placing directly onto embers.
8) Hot smoking : still widely practiced, involves cooking food in the hot gasses from a low fire
9) Air drying : in some environments food can be cured simply by cutting into thin strips and letting it dry in the wind.
Points to note.
In most circumstances the best way to cook it with a fire which is burned down to a good bed of embers with little or no flame or by indirect heat such as heated rocks. This provides a much more controlled and consistent heat than an open flame. In some cases this may require having separate fires for cooking and heat light, or you can always transfer embers from your main fire to a separate cooking area.
Take care when cooking that the fuel and any plant material used for wrapping, skewering food etc is not poisonous.
Some types of rocks can explode when heated, this is a whole separate subject but in general avoid flints or stones taken from a river bed.