Deserializing Json array with variable names first using C# Json.NET

You could make a custom JsonConverter to handle this conversion during deserialization. The conversion code is really not much different than other answers here, except that it is encapsulated into a separate class so that you don't muddy up your main code with the conversion details. From the point of view of your main code it "just works".

Here is how to write the converter:

public class TestClassArrayConverter : JsonConverter 
{
    public override bool CanConvert(Type objectType)
    {
        return (objectType == typeof(TestClass[]));
    }

    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        JArray table = JArray.Load(reader);
        TestClass[] items = new TestClass[table.Count - 1];
        for (int i = 1; i < table.Count; i++)
        {
            JArray row = (JArray)table[i];
            items[i - 1] = new TestClass
            {
                AGE = (string)row[0],
                POP = (string)row[1],
                SEX = (string)row[2],
                DATE = (string)row[3],
                us = (string)row[4]
            };
        }
        return items;
    }

    public override bool CanWrite
    {
        get { return false; }
    }

    public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
    {
        throw new NotImplementedException();
    }
}

And here is how you would use it:

var test2 = JsonConvert.DeserializeObject<TestClass[]>(jsonStr, new TestClassArrayConverter());

Fiddle: https://dotnetfiddle.net/68Q0KT

You have to do the processing on your own, as there is no way the json deserializer can know, how to put the values into the respecitve variables.

If you know, this will be exactly this structure, you could for instance add an appropriate constructor

public TestClass(string[] values) {
    AGE = values[0]; 
    ...
}

to your class. Then serialize your result to array of arrays of string and then pass the inner arrays to your constructor.

var t1 = JsonConvert.DeserializeObject<string[][]>(jsonStr);
//skip the first entry, as this contains the headers
var t2 = t1.Skip(1).Select(x=> new TestClass(x));

If your structure varies, you'll have to write some more complicated mapping code.

You will have to do some custom mapping as your Json does not have any naming conventions so you will have to work with the data in array and index formats. This will work:

var jsonStr = "[[\"AGE\",\"POP\",\"SEX\",\"DATE\",\"us\"], [\"0\",\"3948350\",\"0\",\"7\",\"1\"], [\"1\",\"3962123\",\"0\",\"7\",\"1\"], [\"2\",\"3957772\",\"0\",\"7\",\"1\"], [\"3\",\"4005190\",\"0\",\"7\",\"1\"], [\"4\",\"4003448\",\"0\",\"7\",\"1\"], [\"5\",\"4004858\",\"0\",\"7\",\"1\"], [\"6\",\"4134352\",\"0\",\"7\",\"1\"], [\"7\",\"4154000\",\"0\",\"7\",\"1\"]]";
var test2 = JsonConvert.DeserializeObject<string[][]>(jsonStr);
var test3 = test2.Select(x => new TestClass()
{
    AGE = x[0].ToString(),
    POP = x[1].ToString(),
    SEX = x[2].ToString(),
    DATE = x[3].ToString(),
    us = x[4].ToString()
}).ToList();

//test Case

using Microsoft.VisualStudio.TestTools.UnitTesting;
using System.Collections.Generic;
namespace ApiController.Test
{
    [TestClass]
    public class DownloadIrregularJsonStringObjects
    {
        string ApiKey => "YourPersonalCensusKey";

        /// <summary>
        /// You have to get your own ApiKey from the Census Website
        /// </summary>       
        [TestMethod]
        public void TestGetItem()
        {
        string url = $"http://api.census.gov/data/timeseries/healthins/sahie?get=NIC_PT,NAME,NUI_PT&for=county:*&in=state:*&time=2015&key={YourPersonalCensusKey}";
        string expected = "Autauga County, AL";
        IList<HealthData> actual = ApiController.DownloadIrregularJsonStringObjects.GetCensusHealthData(url);
        Assert.AreEqual(actual[0].NAME, expected);
    }
}
}

///Actual Assembly

using Newtonsoft.Json;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;

namespace ApiController
{
   public  class DownloadIrregularJsonStringObjects
    {
        public static IList<HealthData> GetCensusHealthData(string url)
        {
            var json = GetData(url);
            var rawData = JsonConvert.DeserializeObject<string[][]>(json);

            var headerRow = rawData.First();

            var nic_pt_Index = Array.IndexOf(headerRow, "NIC_PT");
            var name_Index = Array.IndexOf(headerRow, "NAME");
            var nui_pt_Index = Array.IndexOf(headerRow, "NUI_PT");

            IList<HealthData> retVal = new List<HealthData>();

            foreach (var r in rawData.Skip(1))
            {
                HealthData dataRow = new HealthData();
                dataRow.NIC_PT = r[nic_pt_Index];
                dataRow.NAME = r[name_Index];
                dataRow.NUI_PT = r[nui_pt_Index];
                retVal.Add(dataRow);                
            }
            return retVal;
        }

    private static string GetData(string url)
    {
        using (var w = new WebClient())
        {
            var jsonData = string.Empty;
            jsonData = w.DownloadString(url);

            return jsonData;
        }
    }
}
public class HealthData
{
    public string NIC_PT { get; set; }
    public string NAME { get; set; }
    public string NUI_PT { get; set; }       

}
}