You can just set it up on your router. It's hard to give exact
instructions, because there are so many different routers!
You need to set-up port-forwarding, to forward from port 80 on you
router's Internet interface, to port 80 (or whatever port you use
for http) on your development machine.
Note: some Internet providers will block port 80, because they
might not permit you to run a website on your home Internet
connection. You have to pick some different port, then. Port 8080 is
popular! In that case, you do not have to change the port number on
your development computer - just on the Internet side, in order to
work-around the port block.
You have to check with the legalities. But, if you read the find
print, it is usually fine when you are using it for development/test.
They just don't want you running a permanent website and bring
hordes to your Internet connection. And/or nobody is going to notice
this level of activity.
Secondly, it is convenient to sign-up with some dynamic domain
service. You can get these for free or cheap. Your IP address will
probably change periodically, unlike a professional web hosting set-up
where you get a permanent IP. You will install some little plugin or
application, that will keep the service up-to-date with your current
IP address. Or, many/most current routers support many of the popular
services directly, and so you just plug in your service user
ID/password right on your router setup.
You might also get your own domain name, and then you can use it
with the dynamic IP service. Otherwise, you can use a subdomain on the
dynamic service.
P.S. Since it's such a common reason for using a dynamic IP
service, the service will almost certainly have instructions on how to
do the port-forwarding for common routers! And they will surely tell
you how to set-up the update with common routers.