Thank you for the help Felix!
Just connecting the GND, TX and RX worked perfectly.
The camera is however not allocated an IP address as it says it is 0.0.0.0.
I have it connected to an ethernet switch with DHCP enabled. Reading online I see several people having problems using the W5100 ethernet controller with switches. Is this something you are aware of?
Is it possible to use the FTDI adapter to program a static IP to the camera without modifying the adapter or do I need to buy the official downloader board?