To fully customize the Youtube player, you need to make use of Youtube API, and then you would have to program a lot of Javascript code that will be a long and truly extensive process as you can take a look at in this article which explains how this process is done.
Fortunately there is software/plugins already ready for this, which makes life a lot easier, such as 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 first you have to install it in your script, calling it as for example we usually call a jQuery library, only 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: Implementation of the jQuery Library is also required for the plugin to work
Then, to customize the player in your own way, just make the changes in 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 of having a youtube player completely customized in your own way only with the implementation of this Plugin.
Here is an example the end result of how it will look.