Most probably your router is on the same network range as the proxy server. If you have 2 NIC's on your proxy you could configure the one NIC to use the router and the other NIC to your local network.
This is most probably the safest way to do this as your private clients cant access the internet unless via the proxy. Your gateway will then be the proxy server.
On the proxy server you must disallow all other traffic to go to the internet except the allowed protocols. Then all clients that needs to connect to the internet must use <proxy-server
ort> to connect.
If you don't have 2 NICs you might want to setup your router to only allow traffic from the proxy-server and /or allowed protocols nl. 25, 110 and so on. If it is a smart router you can configure it to only allow the proxy-server on port 8080 for instance.
Hope this helps....