First off - always use strictures (use strict; use warnings;).

Perl initializes everything for you (to undef mostly) so if you don't have a real value to assign to a variable at declaration time don't assign anything. Combined with use warnings; that often gives early warning about logic errors.

Declare variables as late as possible so the scope over which they are used is clearer.

Uber points for checking the result of your opens, but you should use the three parameter open for extra safety and clarity.

Don't use & for calling subroutines - it's generally not needed and often doesn't do what you think.

Don't use global variables in subs (askname for example) - it's often unclear where variables are altered if you do that sort of thing leading to difficult to debug problems. Your name fetching loop may be better as:

my $playername;

until ($playername) {
    print "What is your player's name?: ";
    chomp($playername = <STDIN>);
}
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Apart from a reprise of the globals and calling convention issues, clipcalc only processes the first line of the log file. It would be better written as:

while (<LOGFILE>) {
    $clipsobt++ if /\bclip\b/i;
}
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You need to put that in a loop, to read the entire file.

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