You can use the No-IP client on the Raspberry Pi, there is no need to configure DDNS on the router.
Your Raspberry Pi does not need to know the router's IP address, it's your Dynamic DNS provider who needs to know it.
No-IP client sends a web request containing your device ID (credentials) to the DDNS service provider. The request is relayed using your router and as NAT (Network Address Translation) takes place, the Pi's local IP address in the packet is replaced with the router's public one.
The Dynamic DNS provider sees the request as coming from your router's public IP address and updates the record for the device to this address.
Now, to reach a service on your Raspberry Pi using the Dynamic DNS address, other devices (clients) on the internet will first ask the DDNS provider for the IP address for your device and domain, and it will respond with your router's address.
The client will then contact your router and in this stage, the router must forward incoming packets to the Raspberry Pi. That's what you need port forwarding settings for.