Android - How can I display my Android on my PC monitor?

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?

Just to update this answer-- the Chromecast and Nexus player or any Android TV device can now do this for you.

A solution is a dock or MHL (micro-USB form factor) to HDMI adapter. A quick Google or Amazon search should find what you need under $20 USD if you go with a 3rd party and not Samsung.

Note: some ROMs on some models of phones don't support HDMI out (e.g. CM10 on the i9100 I have) probably not be an issue on your S4 but if you've flashed a non-Samsung ROM you might want to double check.