I believe that my

41 $self=shift
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should put the class name in $self.

No - shift (defaults to shift @_) removes the first argument to the subroutine from its argument list (that's @_) and returns it; that's what you are storing into the variable $self. No voodoo here. Saying $self and shift doesn't make an object. The first argument to that subroutine is the string "Entering sub $pkg\:\:$name\n".

Now how do I get the hash value of $self->{logfile} from sub log_message?

A previous ocurrence of that key is in your sub new in LogMessages.pm, but you don't make use of that method. It doesn't make much sense anyways, since all you want is the logfile name, not an object to pass around. So I'd pass that file name when use-ing LogMessages

use LogMessages qw(/path/to/logfile);
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to store that into a lexical hash for the invoking package, in the import() subroutine of LogMessages :

my %logfile;

sub import {
    my $class          = shift;
    $logfile{ +caller} = @_ ? shift : "/tmp/logfile.$$";
};
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Then you get the logfile stored for the using package from the lexical hash:

sub _log : ATTR(CODE) {
    my ($pkg, $sym, $code) = @_;
    if ( DEBUG ) {
        my $name = *{ $sym }{NAME};

        no warnings 'redefine'; # Turn off symbol redefinitions within
+ the block.

        *{ $sym } = sub {
            log_message ($logfile{$pkg}, "Entering sub $pkg\:\:$name\n
+");
            my @ret = $code->( @_ );
            log_message ($logfile{$pkg}, "Leaving sub $pkg\:\:$name\n"
+);
            return @ret;
        };
    }
}
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and then you have the logfile as first argument in the logging sub:

sub log_message {
    my $logfile = shift;
        
    open my $LOGFILE, '>>', $logfile
        or die "Cannot open logfile '$logfile': $!";
    my $time = getLogTime();
    print $LOGFILE $time, " ";
    for (@_){
        print $LOGFILE @_;
    }
}
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Note three argument open and lexical filehandle. Don't forget to put a 1; at the end of your LogMessages.pm ...

You are using @EXPORT and are stuffing Exporter into @ISA, but there's nowhere you do use Exporter in LogMessages. You don't need it, either, since you have nothing to export.

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}