You need to obtain a TK::Font object for the font you wish to use, then call the measure( $text ) method:

#! perl -slw
use strict;
use Tk;
use Tk::Font;

my $mw = MainWindow->new();

for my $name ( 'system' ) { #$mw->fontNames
    print "Using font $name";
    my $font = $mw->fontCreate( $name );

    for my $text (
        'The quick brown fox jumps over the lazy dog',
        'THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG'
    ) {
        printf "$text measures %d pixels\n", $font->measure( $text );
    }
}
__END__
C:\test>junk44
Using font system
The quick brown fox jumps over the lazy dog measures 302 pixels
THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG measures 409 pixels
[download]

See the Tk::Font POD for (a little) more info, then google for more.