For testing you can do the following without spending any money or a learning curve with some Dynamic service.
Open the port on the home router you wish to use, something like 5505
using something like What's My IP Address?
] you can get your IP address or host name. Now as long as your router or line doesn't reboot or reconnect that will be (semi) static.
Armed with these two you now have a way to connect to your home network from the internet.
InetAddress a = InetAddress.getByName("your_host_name_from_the_site_above");
String port = 5505 ;
String ip = a.getHostAddress();
If your router reboots or drops the line you will need to change the host name again.
Some routers allow for dynamic serving/setup, I'm afraid you will have to google how to open ports or dynamic setup for your specific router.