extending wifi signal trough second router (and second computer?)

I tried to search for a solution to my problem, but didn't quite find what i'm looking for. I know i can buy a range extender, but i would of course prefer to save that money if i can do it with what i have...

So this is the story...

I own a bar and offer free wifi hotspot. The bar is on the ground floor, and my apartment is on the first floor. I used to have the router (Thomson speedtouch ST780i WL) in the apartment directly on top of the bar, so i had good signal everywhere - in the bar and on the terrace. However, i had to move the router to another room and now the signal is very scarce, only few spots in the bar get one bar, maximum two... My media computer (which i use to play music and videos from) is in a place where it has descent signal, so i can play internet radio and whatever... I also have another ADSL router (Netgear WGR614) which i replaced with Thomson because it's weaker.

I found online i can connect the two routers with wires and the other one might be able to work as an access point (i can't do that wirelessly, can i?). But it's on another floor, so wires are out of the question...

Now for an idea i had of a solution.
Could i connect the Netgear with cable to my media computer in the bar (which has Thomson's wifi internet access) and set it up as an access point from there? So when people connect, they use the Netgear.

I hope i explaind it ok, and thank you for your time!