Alright, here is what I’m thinking so far:
If you want this to be a website in the strict sense (with a domain like google.com) then running it on your computer is not a viable choice for you.
If you’re going to be the only person using this website then you can follow @wenn‘s advice, but since you can’t use MAMP I recommend using Uniform Server. It lets you run a website from your own computer, which you can open by typing http//localhost/ into your browser of choice.
Uniform server has everything you need to get started – the Apache webserver, PHP and the MySQL database.
In the end though, this is a very technical approach, you’d have to create a whole bunch of code, and I don’t think this is a very good way to attack your problem. Depending on how much you’re willing to learn and how long you’re willing to spend tinkering with something you may consider finding an existing solution. This might sound silly but Facebook already does all of the things you asked for.