View Can You Make A Gif Your Background PNG. You can pick an animated gif in many ways. Export gif's with a transparent background using adobe after effects and photoshop.
If by saying gif you mean moving picture, you can't do it traditionally.
Use the url field to point it to a source for a gif. How to make a gif with photoshop. Upload and edit gifs, share everywhere. Animated gifs tend to get annoying pretty fast and you might be alienating followers who would otherwise come to your twitter page.