I would recomend you to get a programmer to write up a webpage like
system with a backend database that runs on your local computer in the
clinic and can be accessed remotely by other computers(inside your
clinic) as long as it is connected to the local area network (LAN)
either by wired or wireless.
If you wish to connect to the system remotely outside of your clinic,
such as from home, you need to host the system online so you can
access it remotely from home via internet connection.
To get your system online, it is either you run a web server yourself
or you can buy a hosting package to host your system online. Since you
are a doctor, where you keep patients' data is very important. I would
definately not going to the second option, i think the safetest way is
to run the webserver yourself with the help of a network
For security purposes, I would not even recommend you to put the
system online to allow remote access outside the clinic. But if you
really need that, you can put it online and use SSL (Secure Socket
Layer protocol) to secure your data.
May i know which country you reside in? So i could find some reliable
host for you or even programmer for you.