Question: I want to use post to update a database and don't want people doing it manually, i.e., it should only be possible through AJAX in a client. Is there some well known cryptographic trick to use in this scenario?

Say I'm issuing a GET request to insert a new user into my database at `site.com/adduser/<userid>`. Someone could overpopulate my database by issuing fake requests.

Answer: 
There is no way to avoid forged requests in this case, as the client browser already has everything necessary to make the request; it is only a matter of some debugging for a malicious user to figure out how to make arbitrary requests to your backend, and probably even using your own code to make it easier. You don't need "cryptographic tricks", you need only obfuscation, and that will only make forging a bit inconvenient, but still not impossible.
This is some authorization issue: only authorized requests should result in the creation of a new user. So when receiving such a request, your sever needs to check whether it’s from a client that is authorized to create new users.

Now the main issue is how to decide what request is authorized. In most cases, this is done via user roles and/or some ticketing system. With user roles, you’ll have additional problems to solve like user identification and user authentication. But if that is already solved, you can easily map the users onto roles like *Alice is an admin* and *Bob is a regular user* and only admins are authorized to create new users.