The first question that comes to mind, because you do not state this, what computer controls your Wireless Router?
A
home network router usually has at least one PC directly connected via a LAN cable, from the Network Card on the system. At work, our local Server is directly connected to our ISP's router.
You also do not specifically state if the PC in question is also on wireless, but your reference to "signal strength" implies wireless.
Having said that, your problem looks like the PC in question is not setup for
Obtain an IP address automatically AND
Obtain DNS server address automatically.
Your DNS server is your Router.
You may want to contact your ISP for specific help with your router, since the specify what is needed at your end.
Then, your wireless "card" MAY have gone South