I know this works for Blogger --- not so sure about Wordpress or other weblog publishing tools.
First, you need to find an animated GIF. There are a ton of them on the free-to-join, image hosting site - Photobucket
When you find something you like,
you see an email code, a direct code and an image code written underneath. For blogger you need to right click, copy the direct code in the grey box.
Now, go to your post and wherever you want the smiley to appear, you click the photo icon above your post
a box will appear
You need to choose 'From a URL' and paste the code you just copied into the box below. When you click 'add selected' (box at the bottom of the page), the smiley will appear in your post. That's it. I've put some of my favourite animations with their codes below if you want to try them out. They come from my account so they're not dodgy or copyrighted.
and my all time favourite dancing moomoo