The first problem I see is that you increment "$count" as soon as you put the first line of each entry into each of your arrays. Then when you hit the second line of each entry, $count is putting the long string of letters into the next element of your "pdbString" array. So when you go to print to your output file(s), you are getting the wrong pdbString data for each entry (and this string is empty/undefined for the first entry).

(Update: Forgot to mention: because of the way you used your $count variable, all the "pdbString" values end up as empty strings: you increment $count before you store the string in that array, then when you hit the start of the next entry, you assign "" (empty string) to that same array element.)

Here's a cleaned up version of your code, with "use strict" included (because that's what all good monks do), and a fair bit of seemingly unnecessary stuff left out.

#!/usr/bin/perl -w

use strict;

my ( @PDBchain,
     @PDBcode,
     @pdbString,
     @SCOPclass );

open (IN, $ARGV[0] );

while (<IN>)
{
    chomp;
    if ( /^>d(\d\w{3})(\w[_\d])\s(\w)/ ) {
        push @PDBcode, $1; #SAVE PDB CODE
        push @PDBchain, $2; #SAVE PDB CHAIN
        push @SCOPclass, $3; #SAVE SCOP CLASS (A,B,C,D,E,F,G)
        push @pdbString, "";
    }
    else {
        $pdbString[$#pdbString] .= $_;
    }
}
close IN;

print "Total number of entries: ".scalar( @PDBcode ). "\n"; #COUNT TOT
+AL NUMBER OF PDBS REPRESENTED

for ( 0 .. $#PDBcode ) {
    open (OUT, ">>$PDBcode[$_]_$PDBchain[$_].fa");
    print OUT ">",$PDBcode[$_], "_", $PDBchain[$_], "\n";
    print OUT $pdbString[$_], "\n"; 
    close OUT;
}
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Given the input you showed, that produces two output files, with one entry in each file. BTW, the OP code saved stuff into a "@SCOPclass" array, but then never used it. Was that intentional?

Oh, another thing about the OP code: the last two print statements of the while loop are never reached. That's okay, you don't need them anyway, but proper indentation of your code might help to avoid misunderstandings.