Is it true that the first C compiler was written in C itself? Then, how was it executed and compiled? Or, was this compiler written in assembly language?
It is described pretty well in Dennis Ritchie's writeup of the C language history.
Giving just a summary of what he wrote there, use his article for the finer details. C started with the BCPL language, Ken Thomson had access to a compiler for it that ran on their General Electrics 635 main frame. Unhappy with the language, Thomson used BCPL to write a compiler for the B language, an evolutionary step beyond BCPL that removed some of the technical problems in BCPL.
They used B to create programs on their PDP-7 minicomputer, although most of them were little more than toy programs, the machine had very restricted hardware capabilities. One important step taken by Thomson was to rewrite the B compiler in B itself. A common bootstrapping step.
That compiler was then gradually tweaked, through a short-lived intermediary step called NB (New B) to start resembling C. The PDP-11 minicomputer was instrumental in that step, giving them enough room to improve the language and compiler.