If you know how to do it with the file hardcoded, just change that line (probably something like my $file = 'myfile.txt';) to

my $file = $ARGV[0];
[download]

It's far more common to specify non-file arguments first, i.e. to call the script like

script.pl word file1 file2...
[download]

with possibly many files listed (cf. grep). Perl makes this approach easy with the diamond operator:

my $word = shift;
while (<>) { # Reads the files line by line,
             # ... $ARGV contains the filename.
}
[download]

You can use eof to reset the count if you want to output the count by file.

my declares a variable. It's unrelated to the task; it's possible to write the script without using it (which doesn't mean it's a good idea).