Increasing a value (using AJAX?)

Yes, this certainly can be done. Some programming skills will be required and it all depends on how smooth you need it to work.
The easiest option is to add a new field to Songs table and increase its value by one every time user access a designated voting page. Better option is to allow users to vote right from View or List page, in this case you need to use AJAX in order to update votes number behind the scene.
Besides that, you also need to prevent users from being able to vote more than once for any given song meaning you also need to store their IP address in a separate table and prohibit voting for any given song from the same IP addresses. It really gets more and more complicated if you start thinking of how you can build it right.

Thank you for your reply.
If I knew how to do the above, I certainly would.
Unfortunately, I do not have a clue how to achieve that.

I guess you need to hire a programmer to do that. PHPRunner doesn't come with voting functionality. This is something that requires manual coding.