Hi,
The ADM3202 is a TTL to RS232 transmitter IC.
So you need an interface which is RS232 compatible. That´s all.
If your PC still has mature RS232 ports, then you may use the DB9 board and a RS232 cable.
But you may also use a USB to (true) RS232 converter. We can´t see whether the shown USB cable works with TTL levels or RS232 levels.
You need to be sure that there are true RS232 levels.
Note: RS232 lines need to be crossed: Tx of one device needs to go to Rx of the other device and vice versa.
Klaus