You can install the Newtonsoft.Json Nuget package, then use it to access a JSON object like so:
using Newtonsoft.Json;
MessageBox.Show(myJsonObject["result"][0]["id"]);
Assuming your json object is stored in the myJsonObject variable.
To store the json in a JObject (or a class), you can serialise/deserialise it or create a JObject manually:
var myJObject = new JObject(
                                    new JProperty("message", "Customer updated"),
                                    new JProperty("result", 
                                        new JArray(
                                            new JObject(
                                                new JProperty("id", 1),
                                                new JProperty("customer_name", "Andrew"),
                                                new JProperty("customer_lastname", "freeman"),
                                                new JProperty("customer_identity", "12345678")
                                            )   
                                        )
                                    )
                                );
Though it looks like your JSON could be invalid - you have an array within your result array that doesn't have a property name.
You can store the result object directly within the first array, as I have done in my example above, and remove the 1 as you have this in the id field:
{
    "message":"Customer updated",
    "result":[
          {
             "id":1,
             "customer_name":"Andrew",
             "customer_lastname":"freeman",
             "customer_identity":"12345678",
          }
     ]
 }