If I understood what you are asking, you would like browse internet like you would if you were physically connected to your home network.
You are asking for a VPN, but IMHO you should use a Proxy, a service that allows clients (like your browser) to make indirect network connections to other network services. I strongly suggest to look how to install and configure the latter rather than the former for both security and easiness of configuration (read more later).
Long story short, you need 3 things.
The first 2 are a public IP and a way to NAT the public IP port to the server/PC that is running the VPN service (NAT: Network Address Translation). Such NATting more often than not requires a static public IP address and it could be possible that your Internet Service Provider would not provide such a service or make it pay more for it. If your ISP does not provide such services or your home router modem is not capable of such a thing, you could try to piggy back an external cloud server, but it will require a lot more configuring that I won’t write down.
EDIT: dynamic DNS could be a solution in case you can’t have a static IP address, but ATM I don’t know how you should configure it because unluckily I have no experience with it.
The third thing you need is the actual service (VPN or Proxy). As a VPN server, I often use OpenVPN (free and opensource, iirc), unluckily I have very little experience with Proxies, but I know for sure there are many free softwares you can try out.
Whatever is your final choice, do NOT rely on password authentication: try to enable and prefer certificate authentication over password auth, because with a port exposed to internet, anyone in the world can attempt to hack it and once successful, it will result that you performed whatever they are going to do. This is also why I strongly recommend a proxy over a VPN: if your proxy get hacked, they could listen to your traffic and/or browse internet, but your VPN gets hacked, they can very likely do anything in your home network and open any connection to the internet from your home.
So, for the actual steps:
- Check if you can NAT a port from your home network to your public IP. If you can’t or it seems it doesn’t work (e.g. a service that has Plug 'n Play enabled will not be exposed to the internet), ask your ISP for assistance and configuration.
- Install the service and note the port to be NATted on your router/modem
- Access your router modem/router and NAT the local network IP:port to your public IP:port
- Now anyone in the world should be able to access your service on your public IP / port, so be sure to have a strong security (certificate authentication)
EDIT: This guide is for educational purposes only. Please use this information responsibly and legally. I am not responsible for any misuse or illegal activities that may arise from the use of this guide.