Sure, you can easy set your second router up as it's own network.
🙂 If all you would like is a separate network, all you need to do is run an Ethernet cable from one of the numbered, wired Ethernet ports (assuming you have more than one Ethernet port on your modem / router combo device from your ISP) of your modem / router to the WAN plug of the router you wish to create a second network with.
The only thing left to do at that point will be to go through the initial setup for your newly attached router, bearing in mind, this particular configuration will create two isolated networks. Devices that are attached to your modem / router by Ethernet cable or wirelessly, will not readily be able to communicate with devices that are connected to the second router, but all devices should have access to the internet.
Keep in mind, if you have poor wireless reception, changing channels on the router may help, especially since you plan to run two separate wireless stations. Also, the orientation of the antenna in or attached to the router and on your device accessing it should be parallel for best reception. Metal or magnetic devices will likely cause great attenuation or degradation of your wireless signal.
Another alternative, if you are unfamiliar would be, Ethernet Over Power line. This uses a pair of devices which plug into a electrical wall receptacle and extends Ethernet signals through house wiring to any other Ethernet Over Power line device connected to the same house electrical circuit. If an electrical outlet near your current modem / router and an outlet near the computer you are having the poor reception at reside on the same electrical circuit, you could try using a pair of devices such as this:
http://www.newegg.com/Powerline-Networking/SubCategory/ID-294?Order=PRICE
The only caveat to Ethernet Over Power line devices is an inability to predict or guarantee any particular level of performance, and they can not be behind any sort of power strip or surge suppressor. As with wireless devices, mileage varies, and sometimes changing outlets can yield significant performance gains.
Just throwing that out there to show there are alternatives to wireless in some cases.
🙂