UPDATE: This achievement (as well as the entire app) is now discontinued
--> Just wanted to add a tip for anybody a little worried about spamming Facebook <--
Once you have linked Machinima to your Facebook account, go into the privacy settings on Facebook and make it so all Machinima posts are seen by "Only me". That way, all the posts made by this achievement won't be seen by anyone else. Peace of mind :)
Now, as the other guides have said, just go to any video, and once it's playing, bring up the controls, hit the share button. Rinse. Repeat.
Even though they're hidden from everyone, if you don't want those posts on your page, just head on over to Facebook and delete them. No harm done.