Proving that $E=mc^2$

The axioms are simple: 1) intertial reference frames are equivalent-physical laws are the same in all of them. 2) the speed of light is the same in all these frames. From those you derive the Lorentz transformation. This is done in most texts that cover special relativity. Then you discover the energy-momentum four vector, which must have a modulus that is independent of reference frame. You get $E^2-p^2c^2=m_0^2c^4$ for its squared modulus. Evaluating in a convenient frame (where $p=0$), you get $E=m_0c^2$.

The Lorentz transformation needs to be differentiated and then solve for $dv$ Please see the link below.

YouTube video by a man called "Physics Reporter".
Deriving the mass energy equivalence formula

I liked

Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik 18 (13): 639–643

It's the first reference at Mass-energy equivalence. You might also try

Einstein, A. (1961), "Relativity: The Special and the General Theory", Three Rivers Press. 1961. ISBN: 0517025302

(Warning: There are apparently electronic versions of this text that lack illustrations. This will be a significant impediment to understanding whether the coordinate transforms are passive or active.)