Hi LILALA,
Refer the below code.
Code
protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        this.BindGrid();
    }
}
protected void Search(object sender, EventArgs e)
{
    this.BindGrid();
}
private void BindGrid()
{
    string constr = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
    using (SqlConnection con = new SqlConnection(constr))
    {
        using (SqlCommand cmd = new SqlCommand())
        {
            cmd.CommandText = "SELECT ContactName, City, Country FROM Customers WHERE ContactName LIKE '%' + @Search +  '%' OR City LIKE '%' + @Search +  '%' OR Country LIKE '%' + @Search +  '%'";
            cmd.Connection = con;
            cmd.Parameters.AddWithValue("@Search", txtSearch.Text.Trim());
            DataTable dt = new DataTable();
            using (SqlDataAdapter sda = new SqlDataAdapter(cmd))
            {
                sda.Fill(dt);
                gvCustomers.DataSource = dt;
                gvCustomers.DataBind();
            }
        }
    }
}
protected void OnPageIndexChanging(object sender, GridViewPageEventArgs e)
{
    gvCustomers.PageIndex = e.NewPageIndex;
    this.BindGrid();
}
protected void OnRowDataBound(object sender, GridViewRowEventArgs e)
{
    if (e.Row.RowType == DataControlRowType.DataRow)
    {
        e.Row.Cells[0].Text = Regex.Replace(e.Row.Cells[0].Text, txtSearch.Text.Trim(), delegate(Match match)
        {
            return string.Format("<span style = 'background-color:#D9EDF7'>{0}</span>", match.Value);
        }, RegexOptions.IgnoreCase);
        e.Row.Cells[1].Text = Regex.Replace(e.Row.Cells[1].Text, txtSearch.Text.Trim(), delegate(Match match)
        {
            return string.Format("<span style = 'background-color:#D9EDF7'>{0}</span>", match.Value);
        }, RegexOptions.IgnoreCase);
        e.Row.Cells[2].Text = Regex.Replace(e.Row.Cells[2].Text, txtSearch.Text.Trim(), delegate(Match match)
        {
            return string.Format("<span style = 'background-color:#D9EDF7'>{0}</span>", match.Value);
        }, RegexOptions.IgnoreCase);
    }
}
HTML
Search:
<asp:TextBox ID="txtSearch" runat="server" />
<asp:Button Text="Search" runat="server" OnClick="Search" />
<hr />
<asp:GridView ID="gvCustomers" runat="server" AutoGenerateColumns="false" AllowPaging="true"
    OnRowDataBound="OnRowDataBound" OnPageIndexChanging="OnPageIndexChanging">
    <Columns>
        <asp:BoundField HeaderStyle-Width="150px" DataField="ContactName" HeaderText="Contact Name"
            ItemStyle-CssClass="ContactName" HtmlEncode="false" />
        <asp:BoundField HeaderStyle-Width="150px" DataField="City" HeaderText="City" />
        <asp:BoundField HeaderStyle-Width="150px" DataField="Country" HeaderText="Country" />
    </Columns>
</asp:GridView>
Screenshot
