What's the spec of the 2nd hand PC? Operating system (Windows ME/XP?), etc?
As you have had interactive telephone support, can you post what suggestions AOL Tech support made and which you have tried unsuccessfully? Will save us all suggesting the same things.
Click on "Start> Run>
Type in "cmd" (but without the quotation marks)
In the new, black window, type "ipconfig /release" (again without the quotes)
Then type "ipconfig /release"
Followed by "ipconfig /renew"
If your router is up and running amd your PC's wireless network connection is correctly configured, the router should assign an appropriate IP address to your PC now.