Combining URIs and Paths

Answer 1

Add a trailing "/" to apiUri, and remove the leading "/" from method.Path:

        Uri apiUri = new Uri("http://www.r-s.co.uk/eproxy.php/");
        string path = "char/SkillIntraining.xml.aspx";
        Uri uri = new Uri(apiUri, path);
        Console.WriteLine(uri.ToString());

Answer 2

Don't use the Uri object, use a UriBuilder - it copes way better with missing slashes

So

Uri apiUri = new Uri("http://www.r-s.co.uk/eproxy.php");
string methodPath = "/char/SkillIntraining.xml.aspx";

System.UriBuilder uriBuilder = new System.UriBuilder(apiUri);
uriBuilder.Path += methodPath;

Console.WriteLine(uriBuilder.Uri.ToString());

works as expected and produces http://www.r-s.co.uk/eproxy.php/char/SkillIntraining.xml.aspx