Perhaps we need to clarify some terminology. To connect to the console of the router you need something that will provide a serial/com port capability on your computer that will connect to the console of the router. When I got a new laptop that did not have the serial/com/db9 port I got an adaptor (USB on one end and 9 pin on the other end - mine is from Keyspan and I have been happy with it). I am not aware of any way to connect from a PC Ethernet to the router console.
Depending on what the person had done with the router before they sold it to you, you might be able to connect to the router without the console connection. If you connect your PC Ethernet to the router Ethernet (a cross over cable for this connection) and if the router is configured to supply DHCP on the interface, then you could get an IP address and could telnet to the router.
But to telnet you need passwords. And if you do not have passwords then you need to do password recovery. And to do password recovery you need console access.
So bottom line is that you need to find some way to get console access for this router.
As I tried to say before we need to clarify the terms that we use. By definition the access to the console port on an IOS router is for async serial connections. There is not a way to use Ethernet on your PC to connect to the console port of the router.
As I explained, it may be possible to connect from the Ethernet of your PC to the Ethernet of the router. But the connection to the Ethernet of the router is likely to require passwords. And I gather from your post that you do not have passwords for the router. And if you do not have passwords then you need to do password recovery. And password recovery depends on having console access to the router. If your PC does not provide the serial connection, then there are USB to serial (9 pin) adaptors that are available. If you get one of those you should be able to access the console port of the router.
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