The problem is that you have a condenser microphone that does not have a power supply. To make that work with an audio mixer, you need some sort of condenser pre-amp or you need to build a circuit to provide bias voltage to the microphone.
Typically, professional and stage microphones have a small pre-amp built into the mic, which provides bias voltage to the condenser and then filters that back out before going to the sound board. PC microphones, however, don't have this. Instead, the bias circuit is built into the sound card.
So far, I haven't found much in the way of powered adapters to connect PC headset mics to mixing consoles; the easiest and probably smartest thing to do would be to pick up a broadcast headset that uses either a dynamic microphone or a powered condenser mic.
Personally, I use an old TASCAM US-125M as a preamp for my headset, with the output routed to my mixing console. I can then use the same headset for both recording through the mixer and straight to a VoIP application like Skype.
Another possibility is to buy a two-piece condenser mic, such as a tie-clip or lavaliere microphone that has a separate power module. Then just plug your headset mic into the power module.
You can use some wireless kits with a 1/8" microphone input. If those come with a headset or tie-clip mic, you can probably plug your PC headset directly in.
Finally, you can build a circuit. You have to apply power to the tip of the plug and use a capacitor to filter that back out before going to the mixer. Leave the ring disconnected.
Here's an explanation: