The green circle shows where I connected the microphone, red ones are the RCA connectors with a 6.3mm adapter and I also connected my headphones. Once I did all that, I can hear the sound coming from my laptop and also hear my voice when I talk into the microphone, but I am not able to record my voice on the laptop.
Not only are you using the input of the mixer, you are most probably plugging the cable into another output. Laptops generally don't have an input jack socket - the jack socket in most cases is a headphone output. Your connection path should be output of the mixer to input on the laptop, but I am fairly certain you won't find an input socket.
Your best solution is to get a cheap audio interface with a usb connection and plug the usb connector into the laptop, thus creating a separate audio interface. You can then connect this desk to your interface, or you can simply find a cheap desk that has a usb output interface.