Ian Morrison wrote up some nice lectures in the book Lectures on Riemann surfaces,World Scientific publishers, Proceedings of the college of Riemann surfaces in 1987, at the ICTM in Trieste.  They were intended as an informal introduction to the two detailed treatments mentioned below by Mumford (l'Enseignement) and Gieseker (Tata).

There is a nice treatment of the chow coordinates of a projective variety in chapter 1 of the book Basic algebraic geometry by Shafarevich.

There is a good discussion of the existence of the Hilbert scheme in Mumford's book Lectures on curves on an algebraic surface, Annals of math studies #59.

Mumford (notes by Morrison) first wrote up the case of stable curves in Stability of projective Varieties, in l'Enseignement mathematique, 1977, based on an idea of Gieseker.

Then Gieseker himself presented his version at the Tata Institute in Bombay (TIFR), and wrote it up in their series of lectures on mathematics and physics, #69, 1982.

The original presentation of the concept of stable curves, due to Alan Mayer and David Mumford, is in talks by Mayer and Mumford at the Woods Hole conference 1964, available on James Milne's web site at Michigan, or that of roy smith (mathwonk) at University of Georgia.