Unable to connect to internet through Linksys WRT160N router. Can unplug it and internet is fine. Tried power cycling the modem and router. Linksys comes up with all lights flashing and will not continue to boot. It is plugged into APC UPS alond with modem and computer. Tried red reset button on back. Still flashing.

If I understand your problem correctly, you are able to connect to the Internet when you hook the computer directly to the modem, but not when the router is connected to the modem.
If this is the case, then you need to do a power drain on the modem. Power cycling the modem is not enough to clear the NVRAM of the MAC address of your computer. You have to unplug the power cord of the modem for 15 to 20 minutes in order for the NVRAM to reset and allow the modem to recognize the MAC address of the router.
After the modem has been unplugged for 15 to 20 minutes, disconnect the power cable from the router and then connect router to the modem using the Ethernet cable.
Next with the router power cord still disconnected, plug in the power cord of the modem and allow the modem to sync to the network.
Once the modem is finished with its start-up, plug-in the power cord of the router and let it sync to the network.
After the modem has completed its start-up, it should now be connected to the internet.
Good Luck