How can I display my Android on my PC monitor? I have scoured the Internet and come here as my last hope.

I currently have the Galaxy S4, M919v. I own two 24 inch Samsung monitors.

Is there any method for me to display my Android screen onto one of my monitors?

I want to basically play my Android games, using my Android phone, but have them displayed on my monitors. My 24 inch monitor has the same resolution as my phone, so I figure the graphics should be great.

I am trying to find a WIFI/micro-usb cable solution to this problem without buying any types of additional connectors.
You mentioned in your question that you don't want to buy any additional connectors. Any reason in particular? How unwilling are you to spend money? Resolution isn't going to be the problem, it's latency.

If you're absolutely adverse to paying, try running a VNC server on your phone then connect to it using a VNC viewer program on your computer. Laptop Magazine has a very good walkthrough here: How to Control an Android Device From Your PC. They reference VMLite VNC Server which is $9.99 but you can find other VNC server programs that are free (though several don't look terribly trustworthy, IMO). You're getting what you pay for, however; a cheap wireless solution means input lag. Your games are going to be very hard to play and the experience will be generally poor.

If you can manage a few dollars, then an Micro USB to HDMI MHL Adapter (with 5pin to 11pin Micro USB Adapter for your S4 - be sure to read the fine print!) as referenced in gt7599a's answer is the way to go. $15 (ish) plus an HDMI cable gets you a wired solution with very low latency.

If you want something wireless but are also wanting to be able to actually play games, your best bet is an adapter that takes advantage of the S4's WiFi Direct/Miracast capabilities. That means a Samsung WI-FI All-Share Cast Hub ($55), NETGEAR Push2TV ($60), or similar.

How much is it worth to you?