Do you mean like a sprited avatar? If so;
1. Google search for an animated version the sprite you want, if Google doesn't have it (and if it's custom, then it probably won't) then you will need to download GIMP. It is free and you will need it to turn your sprite into an animated sprite.
2. Once you've downloaded GIMP and chosen the sprite/whatever images you want to animate open them in GIMP. Here is an example of something I made in GIMP:
3. Layer each sprite in the correct order on the same white board you start with.
4. Now once they are all in order, right click each frame/layer and select Edit Layer Attributes.
5. Give each layer a Name, example "Buzzy1"
6. Then once you've named it you need to set the animation speed so next to "Buzzy1" type "(500ms)" without the quotation marks of course and a space between 1 and (. You have to do this with each frame.
7. 500ms is a relatively good speed for more animated sprites and you can test it to see what it looks like, but I can't remember what you click to do this.
8. Now go File - Export As and export the image as a .gif to wherever you want to save it.
9. Upload it to Imgur, Postimage etc.
10. Post it in your signature or avatar slot and it should animate.
Good luck and happy animating!