The only way I know how to do it is to copy-and-paste the link.
That is how I did my gif file below in my signature tag. I uploaded it to Photobucket and copy-and-pasted the link. I'm pretty sure YouTube has the same feature.
Be aware that avatars and signatures on this forum only allow for so many bytes. That is why the links are used. Links are many less bytes than the actual gif and/or video file.