Hi alya14,
Check this example. Now please take its reference and correct your code.
Database
I have made use of the following table Customers with the schema as follows.

I have already inserted few records in the table.

You can download the database table SQL by clicking the download link below.
Download SQL file
Model
public class Customer
{
    public int Id { get; set; }
    public string Name { get; set; }
    public string Country { get; set; }
}
Namespaces
using System.Configuration;
using System.Data.SqlClient;
Controller
public class HomeController : Controller
{
    // GET: /Home/
    public ActionResult Index()
    {
        List<Customer> customers = GetCustomers();
        return View(customers);
    }
    private static List<Customer> GetCustomers()
    {
        List<Customer> customers = new List<Customer>();
        string conString = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
        string query = "SELECT * FROM Customers";
        using (SqlConnection con = new SqlConnection(conString))
        {
            using (SqlCommand cmd = new SqlCommand(query))
            {
                cmd.Connection = con;
                con.Open();
                SqlDataReader sdr = cmd.ExecuteReader();
                while (sdr.Read())
                {
                    customers.Add(new Customer
                    {
                        Id = Convert.ToInt32(sdr["CustomerId"]),
                        Name = sdr["Name"].ToString(),
                        Country = sdr["Country"].ToString()
                    });
                }
                con.Close();
            }
        }
        return customers;
    }
    [HttpPost]
    public ActionResult UpdateCustomer(int id, string country)
    {
        SqlConnection con = new SqlConnection(ConfigurationManager.ConnectionStrings["constr"].ConnectionString);
        con.Open();
        SqlCommand cmd = new SqlCommand("SELECT COUNT(*) FROM Customers WHERE CustomerId=@P1", con);
        cmd.Parameters.AddWithValue("@P1", id);
        int sonuc = Convert.ToInt32(cmd.ExecuteScalar());
        con.Close();
        if (sonuc >= 1)
        {
            con.Open();
            cmd = new SqlCommand("UPDATE Customers SET Country=@P2 WHERE CustomerId=@P1", con);
            cmd.Parameters.AddWithValue("@P1", id);
            cmd.Parameters.AddWithValue("@P2", country);
            cmd.ExecuteNonQuery();
            con.Close();
        }
        return new EmptyResult();
    }
}
View
<%@ Page Language="C#" Inherits="System.Web.Mvc.ViewPage<IEnumerable<_ActionLink_Update_Record_MVC.Models.Customer>>" %>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head runat="server">
    <title>Index</title>
    <link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.2/css/bootstrap.min.css" />
    <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
    <script type="text/javascript">
        $(function () {
            $('.Update').on("click", function () {
                var id = $(this).closest('tr').find('td').eq(0).html().trim();
                var country = $(this).closest('tr').find('#txtCountry').val();
                var custromer = {};
                custromer.id = id;
                custromer.country = country;
                $.ajax({
                    type: "POST",
                    url: "/Home/UpdateCustomer",
                    data: custromer,
                    dataType: "json"
                });
                return false;
            });
        });
    </script>
</head>
<body>
    <table id="myTable" class="table table-striped table-bordered table-condensed">
        <thead>
            <tr>
                <th style="width: 2px">Id</th>
                <th>Name</th>
                <th>Country</th>
                <th></th>
            </tr>
        </thead>
        <tbody>
            <% foreach (var customer in Model)
               { %>
            <tr>
                <td><%: customer.Id%></td>
                <td><%: customer.Name%></td>
                <td><%:Html.TextBoxFor(t => customer.Country, new { id = "txtCountry", @class = "form-control", style = "width:150px" })%></td>
                <td><%:Html.ActionLink("UPDATE", "UpdateManageUser", "Home", new { @class = "Update btn btn-primary pull-left" })%></td>
            </tr>
            <% } %>
        </tbody>
    </table>
</body>
</html>
Screenshots

Record in database after update
