An ASHX file is like an ASPX file, but it's a handler. That means it doesn't respond back with HTML by default, and can therefore "handle" otherwise unhandled file types, but it's not necessarily tied to that meaning. In this case, you'll only be presenting the response
position=8°rees=180
...to a posted board and piece. So you don't need HTML, so you want an ASHX.
You can make .ashx files the startup page in your project, just the same as .aspx pages. If I were writing a HelloUser.ashx page, I might set it as the start page, with some parameters passed in as querystrings or something.
You're probably going to want a test harness that posts a board / piece to your service, and that could be any kind of project. Command line program, website, test class run through NUnit, whatever. There's a lot of logic to keep track of beyond the "player" logic.
If you need a more detailed answer than that, SO might not be the place for this question. But I wish you all kinds of luck with this - it's an interesting problem.