You can use dynamic domain name services known as DDNS. While recommendations are considered off topic here, you can search Google for ddns and several options will show up. The second entry (http://www.noip.com/free) is recognized as a DDNS provider. I cannot speak to the service quality. There are other service levels/options/sites available too.
You will be required to install a client that informs the DDNS that you are online at ???.???.???.??? IP address.
This is a simple process. However, please understand that your website will not be available when your system is not on-line and there are security issues that you will need to contend with.
You will want to use a firewall https://wiki.debian.org/DebianFirewall on the computer itself along with anti-virus software such as CLAMAV http://www.clamav.net/index.html. I would also recommend keeping your system up to date as much as possible.
One other option to consider is installing MODSecurity https://www.modsecurity.org/ or other option to protect any installed web application from attack.
CMS software packages can often contain vulnerabilities that can make your system very vulnerable to attack and complete loss. For example, the most hack software on the planet for over a decade is WordPress. It is not that WP is poorly coded, quite the opposite. Vulnerabilities can exist in the CMS, plug-ins, and even themes.
You can check out any known vulnerabilities in software and hardware using the CERT NVD database https://web.nvd.nist.gov/view/vuln/search?execution=e2s1 which is the clearing house for all vulnerabilities.
Once you have signed up for DDNS service, you will want to follow their directions and set any required DNS setting within your domain name registrar so that your site can be found. And please do not forget to install the small code required to update your IP address within the DDNS.
This is not commonly done anymore now that broadband is so prevalent. However, it is a good option for those on the go and those who are running demos onsite.