LWP is your friend:

use LWP::Simple;
my $html = get( 'http://api.clickatell.com/http/sendmsg?api_id=APIID&u
+ser=me&password=password&to
+=mynumber&text=Hello+me' );
[download]

Try that. If it fails, post the error message. (You may need to fiddle with the user agent string sent by the request, but we'll worry about this if you need it.)