Quick google search yeilded this site, seems relatively straight forward to use;
http://www.resizeyourimage.com/
Upload your image to it, resize to 60 pixels by 60 (use the zoom as well to bring it down to correct proportion) download the result
Once you've done that you need somewhere to upload it;
All the cool kids these days appear to be using http://imgur.com/
Upload the file you created and get the link. Should stay live as long as its being viewed, which as an avatar should be fairly often.
Alternatively use a site like photobucket.com (though you have to sign up for that one)