No image formats allow animation besides GIF formats.
The only downside to GIF is that it requires a lower amount of colors and such than JPEG (but if you are making an avatar, it'll be small enough that it doesn't matter).
Anyway, there's a bunch of ways to do it.
For the easiest way, you can use Adobe ImageReady or Macromedia Fireworks that pretty much can put it together for you (such as if you don't have it saved as images frame-by-frame).
If you already have the image exported frame-by-frame, you can use a program I like called '
AniMagic' - even though the program is extremely old, it works very easily.
Just remember to keep your filesize small or it being functional as an avatar is terrible
