It's not quite that simple - you would need a wireless access point that is capable of being either a repeater or a bridge endpoint, and be compatible with your main router. As a repeater, you would access it wirelessly. As an endpoint, it would need to be plugged into a switch, and your upstairs computers would have to be plugged into the switch.
I'm afraid you are going to have to buy *something* - there is no way to use your second router to help you in this scenario without running a cable up from downstairs or using powerline/phoneline networking adapters in lieu of a cable.
Can you put the router in your room upstairs and try to run the downstairs computer wireless?