It's pretty easy actually.
Let's forget about the button click, if the requirement is just to navigate to another page with some unique identification. It can be done through just simple hyperlink/anchor tag.
Let's see, how the code need to be modified (just for reference)-
Gridview:
<asp:templatefield headertext="Action">
<itemtemplate>
<a href="yourSecondPage?id='<%#Eval(" youruniqueid")="" %="">'">Click</a>
</itemtemplate>
</asp:templatefield>
In the second page (in probably the page load event), check if page came with a querystring named "id" and call a method to bind the gridview in the second page.
Something similar to following-
if(!Page.IsPostback)
{
if(Request.QueryString["id"] != null)
{
bindMyGridviewWith80Fields(Request.QueryString["id"]);
}
}
Hope, it helps :)
If there are further doubts or in case I have missed anything here, please let me know.