Do you already know network protocols used to stream media content, like RTSP?
If not, you probably need to start with that.
You should find good code to start with in the Helix source code.
You may face the usual issue of your network mobile operator filtering out anything but HTTP, which is not a very practical protocol for what you want to do.
I wouldn't advise using Java as simply synchronizing audio and video capture will be a nightmare and I wouldn't rely on being able to open a GCF streaming connection.
For Symbian OS C++, I would suggest starting with the multimedia example code from the Quick Recipes book then iterating through the Media Framework plugins installed on your device to see whether one of them can stream video.
If your handset doesn't already have a Media Framework plugin that can do this, you will probably want to give up once you realize that you basically need the handset manufacturer to sign any additional plugin you might want to develop yourself.
At this point, it would be worth going through the Qt documentation and example code since Qt has decent multimedia capability and was first developed with desktop computers in mind.
Video streaming from a mobile phone might also be a bad idea in and of itself, considering the reliance on good network bandwidth to upload lots of data very quickly and the fact that moving the handset while it records video tends to produce bad video quality because of the high video compression used to keep bitrates at a minimum.