TL;DR Don't connect your friend's computer directly to the internet. Please use the service provided by a render farm instead.
It is possible to establish a connection between two computer that are in separate private networks through the internet. However, creating a render farm like this is technically difficult and not advisable for lay-person.
In order to understand why this not easy to achieve, consider the following problems:
Network address translation
Your computer at home is in a private network. In order to communicate with other computers on the internet you need a public IP address. Your internet service provider likely only provides a single public IP address for you and your family. Since you want to connect more than one device at a time to the internet, your router supports Network Address Translation (NAT). Essentially the NAT remembers when you start a connection from a computer from inside your local network to another computer on the internet, e.g. a webserver. When the webserver sends a reply, the NAT knows which computer should get the response. In your example both your computers are behind a NAT, which make establishing a connection difficult. This issue is known as the NAT traversal problem. While not impossible to solve, this may require a proxy/relay server besides your two computers.
Dynamic IP address
Internet service providers (ISP) may assign the public IP addresses to their customers dynamically. This means that the public IP address of your router and that of your friends changes after some some time. Once in a while you need to re-establish the connection. This may affect long running render jobs.
Network Security
Depending on how you intend to establish the connection between the two computers, there may be severe security implications for your friend's computer and network. Please do not persuade your friend to let you configure their firewall, install a remote desktop solution or VPN software on their router. While there may be tempting solutions for your idea, improper configuration could leave your friend's network open to attacks from the internet. Bots regularly scan IP address ranges for vulnerable or badly configured services that respond to their probing requests. Your friend also shouldn't simply hand you the access to their network, since they should be able to decide who gets access and who doesn't. If for some reason they no longer agree with your remote rendering arrangement, they should be able to block your access. Authentication and authorization is required, since only people that have the permission by your friend should be able to render on their computer.
Free Software
A software that solves all the listed problems would have to exist. Based on your comments it would also have to be easy to install and configure. Reading between the lines, it should ideally be free of charge.
Unfortunately trade-offs are necessary, since there is no free software/service that would simply allow to use your friend's computer for rendering. The following solutions exist:
Blender Cloud and Flamenco
Flamenco is an open source software for creating your own render farm. The problem is that setting this up yourself requires a significant know-how and since your two computers are in separate private networks you would also need to rent and host your own server. Luckily this service is already provided by the Blender Cloud and they have a Blender add-on that is easy to install. The downside is that Blender Cloud requires a paid subscription. The upside is that you also get access to tutorials, 3D assets from the open movies, HDRI, textures, characters and more.
Render Farm
Instead of using your friend's computer at all, you can also use the service provided by a render farm. The following render farms support Blender.
Note that I haven't used any of these render farms, therefore I can't say anything about the quality/value of the provided service.
