Hi  itsme,
Refer below sample.
HTML
<div class="dropdown ">
    <button class="dropbtn">
        --Country--</button>
    <div class="dropdown-content" id="drp" runat="server">
        <asp:CheckBox Text="India" runat="server" ID="chkInd" />
        <br />
        <asp:CheckBox Text="United States" runat="server" ID="chkUSA" />
        <br />
        <asp:CheckBox Text="France" runat="server" ID="chkFrance" />
        <br />
        <asp:CheckBox Text="Russia" runat="server" ID="chkRussia" />
        <br />
        <asp:Button Text="Search" runat="server" OnClick="FilterData" />
        <br />
        <asp:GridView runat="server" ID="gvCustomers" />
    </div>
</div>
Namespaces
C#
using System.Data;
using System.Data.SqlClient;
using System.Configuration;
VB.Net
Imports System.Data.SqlClient
Imports System.Data
Code
C#
protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        DataTable dt = GetData("SELECT CustomerId, Name,Country FROM Customers");
        gvCustomers.DataSource = dt;
        gvCustomers.DataBind();
    }
}
private static DataTable GetData(string query)
{
    string constr = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
    SqlConnection con = new SqlConnection(constr);
    SqlCommand cmd = new SqlCommand(query, con);
    SqlDataAdapter da = new SqlDataAdapter(cmd);
    DataTable dt = new DataTable();
    da.Fill(dt);
    return dt;
}
protected void FilterData(object sender, EventArgs e)
{
    string query = "SELECT CustomerId, Name,Country FROM Customers";
    query += " WHERE Country IN (";
    foreach (CheckBox chk in drp.Controls.OfType<CheckBox>())
    {
        if (chk.Checked)
        {
            query += "'" + chk.Text + "'" + ",";
        }
    }
    query = query.Substring(0, query.Length - 1);
    query += ")";
    DataTable dt = GetData(query);
    gvCustomers.DataSource = dt;
    gvCustomers.DataBind();
}
VB.Net
Protected Sub Page_Load(ByVal sender As Object, ByVal e As EventArgs) Handles Me.Load
    If Not Me.IsPostBack Then
        Dim dt As DataTable = GetData("SELECT CustomerId, Name,Country FROM Customers")
        gvCustomers.DataSource = dt
        gvCustomers.DataBind()
    End If
End Sub
Private Shared Function GetData(ByVal query As String) As DataTable
    Dim constr As String = ConfigurationManager.ConnectionStrings("constr").ConnectionString
    Dim con As SqlConnection = New SqlConnection(constr)
    Dim cmd As SqlCommand = New SqlCommand(query, con)
    Dim da As SqlDataAdapter = New SqlDataAdapter(cmd)
    Dim dt As DataTable = New DataTable()
    da.Fill(dt)
    Return dt
End Function
Protected Sub FilterData(ByVal sender As Object, ByVal e As EventArgs)
    Dim query As String = "SELECT CustomerId, Name,Country FROM Customers"
    query += " WHERE Country IN ("
    For Each chk As CheckBox In drp.Controls.OfType(Of CheckBox)()
        If chk.Checked Then
            query += "'" & chk.Text & "'" & ","
        End If
    Next
    query = query.Substring(0, query.Length - 1)
    query += ")"
    Dim dt As DataTable = GetData(query)
    gvCustomers.DataSource = dt
    gvCustomers.DataBind()
End Sub
Screenshot
