Thank for you the clarification.
Here is one possible solution. Hopefully the comments in the code will help.
It uses the JavaScript SDK Version of the Facebook Like Button. You'll need to replace the sample Facebook Like Button code below with your own (and also make sure that the path to your video file is correct in the video tag).
<!-- Wrap video tag and Facebook Like Button in parent container with position relative -->
<div id="wrap" style="position: relative;">
<!-- Video tag -->
<video src="/movie.mp4" width="800" controls>Your browser does not support the video tag.</video>
<!-- Facebook Like Button - JavaScript SDK Version - Code from Step 2 -->
<div id="fb-root"></div>
<script>(function(d, s, id) {
var js, fjs = d.getElementsByTagName(s)[0];
if (d.getElementById(id)) return;
js = d.createElement(s); js.id = id;
js.src = "//connect.facebook.net/en_GB/sdk.js#xfbml=1&version=v2.6";
fjs.parentNode.insertBefore(js, fjs);
}(document, 'script', 'facebook-jssdk'));</script>
<!-- Facebook Like Button - JavaScript SDK Version - Code from Step 3 -->
<div class="fb-like" data-href="https://developers.facebook.com/docs/plugins/" data-layout="standard" data-action="like" data-show-faces="true" data-share="true"></div>
</div>
<!-- jQuery script section -->
<script>
// Execute following when DOM is complete
$(document).ready(function() {
// Show Facebook Like Button only when mouseover video
$('#wrap').hover(function(){
// Mouseenter
$('.fb-like').first().show();
}, function(){
// Mouseleave
$('.fb-like').first().hide();
});
// Initially hide Facebook Like Button, stack it on top of video and position it in corner of video
$('.fb-like').first().css({
'display': 'none',
'position': 'absolute',
'top': '10px',
'left': '10px',
'z-index': '999'
});
});
</script>