I'm afraid I'm not very clear on what equipment you have or how you are trying to set it up.
Your main computer accesses a router by a lan cable. Presumably the router is an adsl wireless router.
You have now fitted a PCI wireless lan card in your spare computer which shows a wireless connection speed of 54Mbit/sec but is not receiving an IP address from the router.
You want to access the internet from your spare computer.
Is this correct?
First, it would help if you tell us the make and model of your router and the operating system you run on the spare computer.
Secondly, set the TCP/IP properties on your spare computer to obtain an IP address and a DNS server address automatically.
Thirdly, log into your route from your main computer and make sure that it is set up as the DHCP server and that it issue at least 2 addresses (it will normally do this by default, but it is best to check). Also, make sure that no wireless security is set at this stage, particularly WEP.
Finally, disable any firewall on your spare pc, at least until you ave established a network connection.