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.
It can be achieved.   

Whenever you render a page which is supposed to make such request. Generate a random token and store it in session (for authenticated user) or database (in case this request is publicly allowed).  

and instead of calling `site.com/adduser/<userid>` call `site.com/adduser/<userid>/<token>`   

whenever you receive such request if the token is valid or not (from session or database)   

In case token is correct, process the request and remove used token from session / db