A GIF is a graphic that may contain motion. In order for a GIF to work.... it can't be an "uploaded" file from your computer. The GIF needs to exist on a web page and be "linked" to.
Put your gif on a webpage (or upload to photobucket or other picture hosting site), then go back to edit your Avatar and instead of "uploading" the image from your computer, use the URL link and enter the URL of where your image is stored.