app/
--app/app.js
--app/form.html

js/
--js/background.js
--js/content.js

var mydiv=document.createElement("div");
    //Next line is where I want to apply some angular functionality
    mydiv.innerHTML += "<div ng-app='app'></div>";

    mydiv.setAttribute('ng-include', "'app/form.html'");
    mydiv.id = "popup";
    mydiv.style.position = "fixed";
    mydiv.style.top = coords.y + "px";
    mydiv.style.left = coords.x + "px";
    mydiv.style.border = "4px solid #d00";
    mydiv.style.background = "#fcc";

    document.body.appendChild(mydiv);
    angular.bootstrap(document.body, ['app'])

<div ng-controller="popupCtrl">
    test
</div>