For avatars, you have some limitations that may be the reason it won't animate here. You're limited to a resolution of 80 x 80 pixels, or a file size of 19.5kb, which ever is smaller.
You must first decide how you're going to host the avatar. You may use something like www.photobucket.com or you may attempt to upload them directly here. Once you change your avatar, it should show.
If you're still having problems with viewing the animated avatars after that, you may need to change your firewall options to allow animation from this site.
This is all assuming, of course, that the avatar is animated in the first place. If it's not, you can use some gif animation software to make it animated, but that is beyond my computer skill level.
EDIT TO ADD: Looking at the properties of your desired avatar, it's resolution is 96x96, so that is probably the reason why it won't work here.
Probably a good thing that they don't work. Lemme tell you, nothing is more annoying than animated avatars/banner ads. If I wanted the internet to dance, I'd wiggle my monitor around.