So, you want the traffic to go other way around. Traffic from the HomeNet should go to the internet via FriendNet, right? In that case, if you want the raspberry box to act as a proxy (or vpn) server, you need to forward relevant ports on the FriendNet to your raspberry pi so that your HomeComputer can connect to the raspberry box.
Or you can set up a VPN and route traffic trough that to the other way. Tunnels work both ways, so it’s possible to set up a route/http proxy/whatever trough the VPN tunnel to the internet, even if the raspberry box is the client from VPN server point of view.
I don’t immediately see the benefit of tunneling your traffic trough the FriendNet to the internet, unless you’re trying to bypass some IP block of something other potentially malicious or at least something being on the gray area. But anyways, you need a method for your proxy client to connect to the proxy server. And in generic consumer space, that needs firewall rules and/or port forwarding (altough both are firewall rules, strictly speaking) so that your proxy server on raspberry box is visible to the internet in the first place.
Once your proxy server is visible to the internet it’s just a matter of writing up few scripts for the server box to send a message to the client end that my public IP is <a.b.c.d> and change proxy client configuration accordingly, but you still need some kind of setup for the HomeNet to receive that, likely a dynds-service and maybe some port forwarding.
Again, I personally would set up something like that with a VPN tunnel from raspberry box to the HomeServer, but as I don’t really undestand what you’re going after with setup like this it’s impossible to suggest anything else.
And then your friend is responsible for your actions on the internet. The end goal you described is so vague that at least I wouldn’t let your raspberry connect on my network.
There’s a ton of VPN services which give you the end result you want without potential liability or other issues for your friend. If you just want to tinker, this thread has quite a bit of information to get you started.