For add new column we can add AutoNumber column by using Container.DataItemIndex property in Gridview like as follows:
<asp:GridView ID="GridViewMyGrid" runat="server" AllowPaging="True" AutoGenerateColumns="False"
PageSize="10">
<Columns>
<asp:TemplateField HeaderText="Serial Number">
<ItemTemplate>
<%# Container.DataItemIndex + 1 %>
</ItemTemplate>
</asp:TemplateField>
<asp:BoundField DataField="Dept_Id" HeaderText="Department Id"/>
<asp:BoundField DataField="Dept_Name" HeaderText="Department Name"/>
//.....
//.....
</Columns>
</asp:GridView>