The main thing you'll want to make sure is that all your JavaScript libraries and images are loading from HTTPS and not HTTP. If you don't, the browser may not load those resources or may show your site as not secured. The default Analytics code will do this for you, but make sure your other libraries are doing so. A great tool for finding any you've missed is Firebug, a Firefox addon. You open it up, load your website and on the Net tab you can see the URL of every resource loaded and even break them down by type (image, JS, etc).
Analytics itself is not affected by HTTPS. Your metrics will continue to load just fine.