Here's a userscript (the relevant part, other than the template wrapper, is literally 7 lines of code by the Power of jQuery™):
// ==UserScript==
// @name Show deleted chat messages for Stack Exchange
// @grant none
// @match *://chat.stackexchange.com/*
// ==/UserScript==
setTimeout(function() {
var userscript = function($) {
$('span.deleted').closest('.message').each(function() {
var id = this.id.replace('message-', ''), _this = this;
$.get('//chat.stackexchange.com/messages/' + id + '/history', function(data) {
var msgtxt = data.match(/<div class="content">([\s\S]+?)<\/div>/)[1].trim();
$('.content > span', _this).html(msgtxt).css({backgroundColor: '#f4eaea', color: '#000'});
});
});
};
var el = document.createElement('script');
el.type = 'text/javascript';
el.text = '(' + userscript + ')(jQuery);';
document.head.appendChild(el);
}, 2000);

To use the userscript, you have a few options:
Install Tampermonkey (Chrome/Chromium) or Greasemonkey (Firefox), and then simply click on this link, which should automagically install the script.
If you don't want to do that, if you're on Chrom{e,ium}, you can still click that link, right click and press "save page as" (or just press Ctrl+S), save it somewhere, type chrome://extensions
into the URL bar, and then drag'n'drop the .user.js file into it, which should also install the script.
For other browsers or alternative options, you can also just Google for "install userscript [browser]."