can you hook up wireless router directly to your pc to connect to internet?

I am in a situation where we have 2 computers in our home. 1 is hooked up to our 50' flat screen tv and has a wireless adapter on back of it. Btw its an hp slimline (win7). and the xbox 360 also has a wireless adapter hooked up in the back of it. I figured hey if the computer hooks up wifi and the xbox is connected wirelessly to that, I wonder if I hook up my wireless linksys router to my computer (9yr old winxp pro gateway performance) directly, it should connect. I tried and I get a limited connection. Anyone know?
 
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