What do I need to build special relativity? If I postulate the principle of relativity and the constancy of the speed of light for every inertial observer can I then prove all SR?
Or do I need some other postulate?
For example: do I need to also postulate the structure of the Lorentz transformations or the Lorentz transformations derive completely only from this two basic postulates. (Do I have to also postulate, for example, that the transformation are linear to prove them from the two starting postulates?)
 A: You can not prove all of SR. You can derive the Lorentz transformation using those two postulates plus linearity. The Lorentz transformation then gives you time dilation, length contraction and relativity of simultaneity. But this is not all of SR. You can not get the relativistic formula for momentum and the well-known formula $E=mc^2$ without also postulating conservation of momentum and energy, and using some definition of momentum and energy.
A: By more modern standards,
if you are looking for a proof
(and not just handwaving arguments),
you'll need to clearly state assumptions about the "space[time]"
and other mathematical structures that model the physics
that you are using as your starting point,
and formulate your postulates in precise terms
with those structures.
(Don't assume that "we all know that THIS [term] means THAT".
Clearly state the assumptions... A successful proof rests on the details.)
You might find enlightening
this diagram describing various pathways
to the Lorentz Transformations. (Sorry I don't have a nicer scan.)


(from "Spacetime and electromagnetism : an essay on the philosophy of the special theory of relativity"
by J R Lucas & P E Hodgson,
Oxford University Press, 1990.)
A: No, the two starting postulates are the only required thing. Lorentz's transformations are also derived from them.
A: Your two postulates would work if you were to build the whole of relativity, including GR. For special Relativity, you would have to include that two objects or frames or reference should be moving without any acceleration and in a straight line.
Once you have the constancy of light postulate, you find out that Galilean transformations do not work. So, you need to derive up a new type of transformations - the Lorentz one; which can bed one by considering some symmetries and space-time diagrams.
