Read the entire fire into a variable. Rather than using fopen you might want to consider file_get_contents for that, it's a bit cleaner.
You'll then want to parse that string as HTML. You could use PHP's DOMDocument for that. Get the action and method of the form by traversing the DOM tree to the form tag and reading out those attributes. Next get the names of any inputs within the form tags. Use those names to generate a query string with your key=value pairs. If the method of the form is GET, then append that query string to the form action, otherwise save it in another variable.
Finally, use CURL to "submit" the form. That is, use the form action as the URL for a CURL request. If the form method was GET, you should have already appended the data to the URL, if the method was POST, you'll want to set the data for the CURL request to the data query string you generated from the form names.
If your question extend to how to know what data to fill into what form fields, that is pretty much impossible to solve. Certainly there are some input names you could look for and guess the required data but a universal solution is an impossible problem to solve.