Your question was
I want my homemade arduino to communicate with my PC. Do I need a max32 chip in this diagram?
The answer is no, you don't need a MAX232 chip, these days you would use an FTDI232 chip and talk using USB.
I have max232 chips & DB9 socket pair in my inventory. So I want to make some useful things out of it.
Now you are saying that you have these components and want to find a use for them, that's a different issue.
Yes you could connect them to an Arduino and talk to a PC, but do you actually have a computer with an RS-232 port? They are few and far between these days.
If you don't and you really want to use the chip then you will have to find something that talks RS-232 and connect to that.
So what is the end goal, to talk to a PC, to use RS-232 with anything, or to talk to a PC using RS-232?