Step 1: Use a site like GIFSoup.com to extract the full video and turn it into a .gif file. (Sure, there are other ways to do it, but for your needs this sort of external service would probably be simplest.)
Step 2: Use some program to select the frames you want (in other words, if you want a three-second clip instead of a ten-minute film, scrap the stuff you don't want) and to crop out parts of the image you don't want. This may be the trickiest part, as a lot of these programs cost money. I use Animation Shop for things like this (free trial here); you might also be able to get by using an online service like Online Image Editor or freeware such as Irfanview.
Step 3: Upload it... somewhere. I use Imageshack for most things, even though their interface has become increasingly convoluted over the years. If you don't like that one, there are scores of other image hosts online, so just find one that suits you.