You should 'use warnings' and 'use strict' in your scripts; they'll help you diagnose a lot of problems.

@logfile = split(/ /, <LOGFILE>);
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The problem with the above is that it's going to treat the <LOGFILE> filehandle as a scalar - that is, it's going to extract one line from it - which is presumably the cause of many of your problems. Try this:

#!/usr/bin/perl -w
use strict;

cd "C:\\Documents and Settings\\Alex\\Desktop";
open LOGFILE, "doomlog.txt" or die "doomlog.txt: $!\n";
open LOGWRITE,">explog.txt" or die "explog.txt: $!\n";

my $clipsobt;
for my $logfile (<LOGFILE>){
    if ($logfile =~ /clip/i) {
        $clipsobt++
    }
}
print LOGWRITE "You have obtained $clipsobt clips.\n";
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Or you can go with a shorter version:

#!/usr/bin/perl -w
use strict;

cd "C:\\Documents and Settings\\Alex\\Desktop";
open LOGFILE, "doomlog.txt" or die "doomlog.txt: $!\n";
open LOGWRITE,">explog.txt" or die "explog.txt: $!\n";

my $clipsobt = grep /clip/, <LOGFILE>;
print LOGWRITE "You have obtained $clipsobt clips.\n";
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Update: Darn it, the node got re?aped while I was answering it (I think.) ++ to GrandFather for reading my brain and typing out the answer for me - wish I'd known and saved myself all the typing... :)

-- 
Human history becomes more and more a race between education and catastrophe. -- HG Wells