There are two principal techniques for making stone tools, depending on the properties of the material you have available.
The first and probably best known is knapping. This is used with glassy rocks like flint and obsidian which can form sharp fracture surfaces when struck or pressed. This process takes a fair degree of skill as you need to understand how the stone will fracture and is a very different technique to conventional stone carving.
Flint tools can perform surprisingly well, in particular they can take an extremely sharp edge, in some cases sharper than can be achieved with steel. The downside is that the material is quite brittle and the edge dulls fairly quickly when used for chopping.
The second method is to shape stone by abrasion as in this example using a second stone as a grinding surface, possibly aided with a sand and water mixture. As an aside it is also interesting to note that a lot of early cast metal axes (ie copper and bronze) look very similar to this type of stone axe.
Having said that in a survival situation making a stone axe is likely to be a very low priority. Building shelter etc only requires thin poles which can usually be cut with even quite a small knife with relatively little effort and standing wood does not usually make good fuel, you are much better off collecting dead wood which should by dry enough to snap easily.
Equally the techniques required are somewhat specialist and labour intensive and so unless you have prior experience there is a good chance that making them will require more effort than is saved.
In terms of binding a common material is rawhide or processed sinew which is applied wet and shrinks as it dries, tightening the bindings. For obvious reasons fixing a stone head to a haft is a bit more tricky than a steel one and it is far from easy to achieve a connection strong enough to withstand the rigours of chopping and you also have to consider the potential danger if the head does come off during use. With all this in mind, in a survival situation just using the head as a hand axe is potentially more practical.