Are you sure you need fork for this? Why not just do something like:

#!/usr/bin/perl
# Pseudo-Perl; some code shortened to ellipses or text.
use IO::Socket;

my $sock = IO::Socket::INET->new(...) or die "...";

while (not yet done) # Not sure how you want to determine this
{
    my $line;
    print $sock "send this to server\n";
    chomp($line = <$sock>);
    #<process $line (one line read from server's response)>
}
__END__
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Also note that

$fork_errlvl = 1 unless defined $pid;
die "Error forking client: Reason: $!\n" if $fork_errlvl != 0;
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can be shortened to

die "..." unless defined $pid;
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or, better IMHO,

defined($pid) or die "..."
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both of which eliminate the unnecessary variable.