The easy way is to use a Flash/Flex Client with Red5 http://osflash.org/red5
The Flash Player has a method for using a video camera and the Red5 server is an open source Flash server that will record client streams.
I recommend setting up Red5 and playing with it. It does everything you need. Just look over the API and start writing test apps.
HOW TO GET VIDEO FROM USER CAMERA:
package {
    import flash.display.Sprite;
    import flash.media.Camera;
    import flash.media.Video;
    import flash.text.TextField;
    import flash.text.TextFieldAutoSize;
    import flash.utils.Timer;
    import flash.events.TimerEvent;
    import flash.events.StatusEvent;
    import flash.events.MouseEvent;
    import flash.system.SecurityPanel;
    import flash.system.Security;
    public class Camera_getCameraExample extends Sprite {
        private var myTextField:TextField;
        private var cam:Camera;
        private var t:Timer = new Timer(1000);
        public function Camera_getCameraExample() {
            myTextField = new TextField();
            myTextField.x = 10;
            myTextField.y = 10;
            myTextField.background = true;
            myTextField.selectable = false;
            myTextField.autoSize = TextFieldAutoSize.LEFT;    
            cam = Camera.getCamera();
            if (!cam) {
                myTextField.text = "No camera is installed.";
            } else if (cam.muted) {
                myTextField.text = "To enable the use of the camera,\n"
                                 + "please click on this text field.\n" 
                                 + "When the Flash Player Settings dialog appears,\n"
                                 + "make sure to select the Allow radio button\n" 
                                 + "to grant access to your camera.";
                myTextField.addEventListener(MouseEvent.CLICK, clickHandler);
            }else {
                myTextField.text = "Connecting";
                connectCamera(); 
            }
            addChild(myTextField);
            t.addEventListener(TimerEvent.TIMER, timerHandler);
        }
        private function clickHandler(e:MouseEvent):void {
            Security.showSettings(SecurityPanel.PRIVACY);
            cam.addEventListener(StatusEvent.STATUS, statusHandler);
            myTextField.removeEventListener(MouseEvent.CLICK, clickHandler);
        }
        private function statusHandler(event:StatusEvent):void {
            if (event.code == "Camera.Unmuted") {
                connectCamera(); 
                cam.removeEventListener(StatusEvent.STATUS, statusHandler);
            }
        }
        private function connectCamera():void {
                var vid:Video = new Video(cam.width, cam.height);
                vid.x = 10;
                vid.y = 10;
                vid.attachCamera(cam);
                addChild(vid);    
                t.start();
        }
        private function timerHandler(event:TimerEvent):void {
            myTextField.y = cam.height + 20;
            myTextField.text = "";
            myTextField.appendText("bandwidth: " + cam.bandwidth + "\n");
            myTextField.appendText("currentFPS: " + Math.round(cam.currentFPS) + "\n");
            myTextField.appendText("fps: " + cam.fps + "\n");
            myTextField.appendText("keyFrameInterval: " + cam.keyFrameInterval + "\n");
        }
    }
}
HOW TO SEND VIDEO TO BRIGHT COVE
They have an API just read over it.