My API as below:

// POST api/values
public void Post([FromBody]string value)
{
}

Calling from C# server side

            string URL = "http://localhost:xxxxx/api/values";
            HttpWebRequest request = (HttpWebRequest)WebRequest.Create(URL);
            request.Method = "POST";
            request.ContentType = "application/json";
            string data = Newtonsoft.Json.JsonConvert.SerializeObject("your body parameter value");
            request.ContentLength = data.Length;
            StreamWriter requestWriter = new StreamWriter(request.GetRequestStream(), System.Text.Encoding.ASCII);
            requestWriter.Write(data);
            requestWriter.Close();

            try
            {
                WebResponse webResponse = request.GetResponse();
                Stream webStream = webResponse.GetResponseStream();
                StreamReader responseReader = new StreamReader(webStream);
                string response = responseReader.ReadToEnd();

                responseReader.Close();
            }
            catch
            {

            }

My API as below:

// DELETE api/values
public void Delete([FromBody]string value)
{
}

Calling from C# server side

            string URL = "http://localhost:xxxxx/api/values";
            HttpWebRequest request = (HttpWebRequest)WebRequest.Create(URL);
            request.Method = "DELETE";
            request.ContentType = "application/json";
            string data = Newtonsoft.Json.JsonConvert.SerializeObject("your body parameter value");
            request.ContentLength = data.Length;
            StreamWriter requestWriter = new StreamWriter(request.GetRequestStream(), System.Text.Encoding.ASCII);
            requestWriter.Write(data);
            requestWriter.Close();

            try
            {
                WebResponse webResponse = request.GetResponse();
                Stream webStream = webResponse.GetResponseStream();
                StreamReader responseReader = new StreamReader(webStream);
                string response = responseReader.ReadToEnd();

                responseReader.Close();
            }
            catch
            {

            }