To fully customize the Youtube player, you need to make use of the Youtube API , and then you would have to program enough Javascript code that will be a long and truly extensive process as you can take a look in this article that explains how this process is done.
Fortunately there are software / plugins already ready for this, which makes life much easier, for example plugins like: PragmaticPlayerJs
This plugin allows you to use a custom player just by adding the following code below:
<div id="youtube-video"><!-- --></div>
<script>
$("#youtube-video").pPlayer({
youtubeVideoId: "YE7VzlLtp-4",
autoplay: 0,
origin: "http://yoursite.com"
});
</script>
But of course you first have to install it in your script, calling it like for example we usually call a jQuery library, except in this case with the Plugin files. Example:
<link rel="stylesheet" type="text/css" href="../pplayer/assets/pplayer.css" />
<script type="text/javascript" src="../pplayer/js/jquery.pplayer.js"></script>
<!-- etc... -->
Note: It is also necessary to implement the jQuery Library for the plugin to work
Then, to customize the player in your own way, simply make changes to the CSS code of the Plugin and if you want, create your own icons and replace the respective images with your new custom icons, thus giving you the possibility to have a completely customized youtube player your way with just the implementation of this Plugin.
Here is an example of the end result of how it will look.