So, I have 1 router setup, it's giving off a wireless signal, but it also has 3 ethernet ports in the back.

Can I take another wireless router and hook it up through the router I currently have setup and give off another wireless signal, using the original router's internet connection?

 Yes, but it very I mean VERY difficult to setup. I tried this. It is the pain in the ass to fix, and sometimes it WILL stop working and there is no solution.Save youseld from this. It is possible but not reccomended.

If all's you want is to broaden your signal range than your best bet is to hook up a wireless access point to 1 of the unused ports. they have a better range and reliability anyway.

 What made it so difficult? Doesn't the second router just take the feed from the first router as an internet connection just as a computer would?

 Nelson,

The problem I have is that I don't really have access to the first router, atleast it's configurations, and it's not secured. What I want to do is take the feed from the first router and connect my own router and secure/configure that router to my liking. So, that's why I am somewhat limited to how I can do this.

It's possible to do that, but the unsecured first router is going to cause you problems as #1 it's unsecured, and #2 if you don't know the configuration settings then setting up the correct IP address on the second router will prove problematic.  It's very easy to get into an unsecure router. 

 Bring up a command console on one of the computers that is attached (wireless or not), you can do this by typing cmd (windows xp or newer), or command (wndows 2k and older) in a run box.  Once the console is up type in ipconfig which will give you...

 IP address:  x.x.x.x

subnet mask: x.x.x.x

gateyway: x.x.x.x

The gateway in these numbers is nearly always the IP address of the router, such as 192.168.1.1.  Next all you have to do is bring up a web browser such as Internet Explorer, type in the gateway address (192.168.1.1 etc..) and then enter what is normally a default username and password.  Most support sites have the default username and password available in their documentation.

 Once you have the first router configured properly you should have no problems with getting the second router in place.

Some routers have a range extender or bridge mode available in their web interface. with those you can connect them to another wireless router wirelessly then connect to the extender/bridge and have the internet pass through the original router. some companies even make special units without the WAN port called range extenders which do this.

 Routers arnt made to be hooked up to other routers.  I believe this causes internet to go slower for me, and just not reliable. I never pin pointed the reason it stopped working.