It is very easy for someone to watch you whilst playing on your computer. I don't exactly know what red dot your on about as my PC has a webcam but has no red dot near it. Red dots near cameras are normally a good indication that the camera is switched on in some way and recording.
Why it is recording though is another question - do you have anything that could be using it running like Skype maybe or some kind of instant messaging program. The other thing is that if your playing a computer game some games do have a "broadcast mode" especially if your playing online with multiple people, or if your playing a game you've bought through, or that runs through an online gaming service like if your playing it through Steam they have a broadcast mode too (you can access these option normally by pressing Shift+Tab in the game to get the Steam overlay up), but normally these modes need to be activated in some way.
Viruses and malware can access resources on your computer totally without your permission. One of the most basic ways for them to do this is to install a program called a VNC server on your computer. This allows them to remotely login to your PC and see your screen, and control your mouse/keyboard, many VNC servers have a password added to them to stop any random person using them, and things like ShowMyPC change the password every time the program is activated, but if the malware contains a VNC server within it then there would be no reason for them to add password authentication into it, or if they did it would probably be one they know. Once they're logged into your computer using VNC they can control your PC by doing things like opening applications up for example the camera program, and then they can see your desktop with the camera program open (and of course what ever your camera is looking at!), but that is a bit obvious though as that would also come up on your screen. The other thing they can do is install a hidden piece of software that tells the camera to automatically stream the video onto a server somewhere (if you've ever seen some webcams on websites this is how they work), that way they could watch it without you knowing at all.
One thing you could try which might give you some indication is, if the red light is on and your in a game press Alt+Tab to leave the game and see if the red light is still lit up, if it is try running your camera software and getting it to capture some video. If your camera is in use then Windows should come up with an error message stating that it can't use the camera as it's already in use (most of the time only one piece of software can grab data from the camera at one time, but there are ways around this). Obviously if you get such a message then you know for certain that something is controlling the camera, if you don't get that message that doesn't necessarily mean your in the clear as there are ways to allow multiple programs to use the same camera, but at least if you do get a message you know for certain that something is playing with the camera.
Your best bet then is to use something like ESET Online Scanner
http://www.eset.com/us/online-scanner
and scan your computer (ESET is probably one of the best for removing all nasties that are hiding) - when using the online scanner go into the advanced settings and tick the boxes
Scan Archives,
Scan for potentially unwanted applications and
Scan for potentially unsafe applications
Enable anti-stealth technology
(as in this picture - http://screenshots.en.sftcdn.net/en/scrn/328000/328664/2011-09-01-02-640x535.png)
Run this and leave the scanner running until it gets to the end - this btw could take quite a long time - possibly best if you do this at night, as we've told it to scan archives as well this can take quite long as it needs to go into each archive in your PC and check it for viruses, or other nasties. This also scans for anything that isn't necessarily a virus, but could be an unwanted or unsafe application, possibly the thing that is activating your camera is more an "unsafe" application than a virus as it might actually be performing something that some people might want.
Once that's done then it should have cleaned out everything and hopefully if it was being activated by some kind of spyware that'll put the end to their little game!