Advice needed: am I going in the right direction?

Your PC may even have an optical out, which I think is better than USB- though some may disagree.
But, big greg is correct, use a USB or Optical cable to the DAC of your choice. The rest is like the usual stereo connections.
As for a sound card, I wouldn't bother. Just use the PC to DAC.
There is a program that doesn't cost much called Fidelizer. It really made a difference for the better.
For a good inexpensive DAC, I would get a Schiit Multibit or Gungnir.
They are available used at a great price, too. Just make sure it is using the current most current updates. But, even if it isn't you can send it in to be upgraded. Great company.
Bob

I would get an Audioquest Dragonfly.  It is a DAC that looks like a usb stick and fits your budget.  You will need a cable that has a mini jack to fit into the DAC and terminates in RCA inputs for your amp

You are correct. Connect computer to DAC, DAC to amp, amp to speakers. Computer needs to recognize DAC, may need a driver depending on DAC and type computer. 

If using USB out from computer and you want to keep it simple and inexpensive there are HRT Music Streamer DACs for sale on Ebay, actually one listed for $57($5 shipping) that would get you started. I have used these in several budget systems and they worked well. Another option is the Schiit Audio Modi 3 which would provide more input options for around $100 shipped.

Good luck 

Thank you so much, guys. 

One thing that is now not that clear for me, is there any difference between a cheap DAC and an expensive one, except for some noise that could be present in a low-priced ones and amount of outputs? Will they produce the same signal for the amplifier?

I'm an amateur musician myself and also wanted to buy the Steinberg UR12 external sound card to make some recordings on it. But will it also work for converting digital to analog for my sound system as an external DAC and will there be a difference in sound between Steinberg and some $100 DAC?