Out of the box, that doesn't work. PC#1 only knows the routes it is told by the upstream router (10.0.0.1 in that setup), and that router doesn't know that unless told. That is akin to the devices on your home network being able to reach out to the internet, but not vice versa (unless you make it so - you generally don't want that...).
Most routers (I have a Netgear R6700) allow you to add custom routes (on that particular router it is under advanced setup -> static routes).
You would add a route to network 192.168.4.0 netmask 255.255.255.0 via 10.0.0.2.
There is a usually an option to keep that route private (meaning only visible from your LAN, not the world). Be sure to check that.
Give it a shot.