C, 306 328 585 611 615 623 673 707 bytes

Source code:

p[256][256]={0};char*X="p[256][256]={0};char*X=%c%s%c,Y[999],c,j,*a;main(){sprintf(Y,X,34,X,34);for(a=Y;*a;a++)p[*a][*(a+1)]++;for(j=*Y;putchar(c=j);)while(p[c][++j]<<16<rand());}",Y[999],c,j,*a;main(){sprintf(Y,X,34,X,34);for(a=Y;*a;a++)p[*a][*(a+1)]++;for(j=*Y;putchar(c=j);)while(p[c][++j]<<16<rand());}

With newlines and whitespace added for legibility/explanation:

01  p[256][256]={0};
02  char*X="p[256][256]={0};char*X=%c%s%c,Y[999],c,j,*a;main(){sprintf(Y,X,34,X,34);for(a=Y;*a;a++)p[*a][*(a+1)]++;for(j=*Y;putchar(c=j);)while(p[c][++j]<<16<rand());}",
03  Y[999],c,j,*a;
04  main(){
05      sprintf(Y,X,34,X,34);
06      for(a=Y;*a;a++)p[*a][*(a+1)]++;
07      for(j=*Y;putchar(c=j);)
08          while(p[c][++j]<<16<rand());
09  }

Explanation

Line 01: p[][] holds the counts of one character following another.

Line 02: X contains the program's source, escaped with %c%s%c.

Line 03: Y will contain the program's literal source. c, j, *a are counting variables.

Line 05: Set Y to contain the quine.

Line 06: Count letter occurrences in p[][].

Line 07: Print the current state.

Line 08: Find the next character randomly, proportional to the counts in p[][].

Sample output:

p[++);p[99]=Y;putfor(aind(a++j,*a+j=j,c][c,*an(arile(pr*Y,Y[256]<<1);)][*Y,Y;)wha+++j=*aintfor*Y;prin(a+j]=j][256<1)pr(a;a;f(p[char(Y;for());};a;ma;ma=%s%chain(Y;ar(j][256<<<1)p[256<<raile(cha][9]<rin(j,34,34,Y[256]+j,Y,34,Y,c=Y,*a;*a;for(){0}