I'd recommend not using the web.config method. customErrors redirects to the error page, which makes little sense. Essentially it first says "oh yes, that'll work perfectly, you just need to go here instead", and then says "oh, we didn't find that". That's really a bug (if there isn't anything here, then why did the server tell me to go here, clearly to the user code it looks like you the server code messed up; they went to the right URI and then you directed them to the wrong one).
Use Server.Transfer() from global.asax, set a default HTTPHandler, or set IIS to execute (not redirect to) your .aspx or other file with your implementation. If you want the same handler to manage each error, then you could, for example, do a Server.Transfer() from global.asax, but include a query string parameter about the type of error (whether simply an HTTP status code, or something more detailed), or pass information in the HttpContext.