I assume you are trying to use the Composite RCA video output jack on the back. My IIc is the same. No Video out via that port. People have told me I need to apply solder to that connector as a potential fix, but because I have Plamen's (A2Heaven)
IIc-to-VGA adapter, I've not set out to do that yet. The Plamen adapter allows me to connect to either a VGA display or RCA/Composite (one that actually works). Rob Kim makes a similar adapter, but one with VGA and HDMI (no RCA) outputs. It's called the
TangyPack.
The following section of 8-Bit Guy's video shows the solder fix for the RCA/Composite plug:
If solder doesn't work, and if the connector itself is smooth and you are certain it's not a connector connectivity issue, then the only other explanation would be a hardware problem. But it is likely one of the two adapters I mentioned would be a workaround for that (possibly).