I didn't. In the end I left the router settings as default, since I couldn't set Pi-hole as the DNS server (before I made it work by setting my router as the upstream DNS), so setting it there was ... yeah, it just brought the whole network down. So I left it as-is and now if I want to manually set Pi-hole as my router's DNS, it makes an endless loop and the whole system gets down again.
Unfortunately my DHCP settings are pretty limited. I basically consists of my router's LAN IP address (192.168.8.1), a toggle to turn on (or off) the DHCP server, DHCP IP range and DHCP lease time (by default set to 1 day). I cannot find an option to distribute Pi-hole via DHCP this way.
I will probably have to experiment with this and see whether this would work ...