http://windows.microsoft.com/en-us/windows/using-internet-connection-sharing#1TC=windows-7
Enabling ICS
To enable ICS, on your host computer:
Open Network Connections by clicking the Start button Picture of the Start button, and then clicking Control Panel. In the search box,
type adapter, and then, under Network and Sharing Center, click View
network connections.
Right-click the connection that you want to share, and then click Properties. If you're prompted for an administrator password or confirmation, type the password or
provide confirmation.
Click the Sharing tab, and then select the Allow other network users to connect through this computer’s Internet connection check box.
Notes
The Sharing tab won't be available if you have only one network
connection.
You can choose whether to select the Allow other network users to
control or disable the shared Internet connection check box.
Optionally, to allow other network users to use services running on
your network, click Settings, and then select the services you want to
allow.
When you enable ICS, your local area network (LAN) connection gets a
new static IP address and configuration, so you'll need to reestablish
any TCP/IP connections between your host computer and the other
network computers.
To test your network and Internet connection, see if you can share
files between computers and make sure each computer can reach a
website.
This allows you to share the connection from one network with another network. If the connection from the first PC to the Access point is using ethernet, that's all there is.
If you want to create a new wireless network that's hosted by the PC and provides internet access to a wireless access point, you need to host a wireless network from the command line, see: Tethering on Windows 7?
From admin Command Prompt type:
netsh wlan set hostednetwork mode=allow ssid=mywifinetwork key=mysecretkey
Enable new virtual Wi-Fi adapter from the Network and Sharing Center (the adapter might be disabled upon creation)
From admin command prompt type:
netsh wlan start hostednetwork
Depending on your network card, this may or may not work. Also, it's entirely possible that the old router doesn't allow you to properly configure it as an access point - ideally you first test that you can connect to the PC's WiFi from your phone, and can use the internet. Sometimes old routers don't allow decent encryption protocols when used as routers. In that case you may want to replace the linksys firmware with something more capable like DD WRT or Tomato. Be advised this is only for advanced users.
If you don't love tinkering, best buy a 5$ USB WiFi card for the old PC.