You want to use a while loop,
#!/usr/local/bin/perl
open (TEXT,"general.txt") || die "Could not open file: $! \n";
$addthis=join("", <TEXT>);
close (TEXT);
open (FILES,"files.txt") || die "Could not open file: $! \n";
# Start to loop here??? or after the next group of statements?
while ($line=<FILES>){
chomp $line;
open (DATA,"$line") || die "Could not open file: $! \n";
$tothis=join("", <DATA>);
close (DATA);
$newrecord=$addthis.$tothis;
open (DATA,">$line") || die "Could not write to: $! \n";
print "$line\n";
print DATA $newrecord;
close (DATA);
}