Adding Custom Field to Contact Membership Tab
The template you're looking for is templates/CRM/Member/Page/Tab.tpl.  The "browse" action is number 16, and takes up the vast majority of the template, starting around line 33.

You'll want to populate the $activeMembers and/or $inActiveMembers array with modified data.  Those are defined in CRM_Member_Page_Tab around line 211, but most likely you'll want to modify the $membership array that they're both based on.  The browse() method of that function is mostly a "while" loop devoted to building the $membership array, but is sadly devoid of hooks.