The problem you are having is likely because the Xbox 360 controller microphone is only used for online chat. Because of this, it appears the console does not send it regular sound. If you connect both the RCA and HDMI at the same time, however, you should be able to use the HDMI for your monitor, and use the RCA audio to connect to your headset.
Connecting HDMI and RCA at the same time
AFAIK, the Xbox 360 RCA plug blocks off HDMI, but if you really want to get it connected and feel confident enough, there is a good guide to modify the RCA plug to accommodate the HDMI. You basically use a thin flat-head screwdriver to pop open the plastic surrounding the plug connector, and remove the side facing the HDMI, before insulating it with electrical tape.
Connecting the headset
Without knowing the adapter you are using, specifically, we can at least assume that you are at the stage where the headset is working with the standard Xbox 360 audio port, 2.5mm.
What you need is a 2.5mm female to 2 RCA female adapter. It might be easier to use a 2.5mm to 3.5mm adapter, and a 3.3mm female to 2 RCA female adapter, as 3.5mm can be a lot more common. If you do not know what any of this is, go into your local electric wholesaler. Places like ebay offer a cheap alternative, too.
Just connect everything up, and you should get sound through your headset, and video through your monitor.