I don't think you have DVT, but do you only sit when you play videogames or do you also sit for hours at school or work? If it's just 2-4 per day and you're fairly active, you should be fine. If it's a lot more, then try to get up and move around when playing.
That said, DVT often comes with other symptoms - a warm, reddish swelling (localized or otherwise) of a leg, leg pain, etc. But it normally wouldn't last a month. It doesn't sound like an emergency, but it might be worth going in for a regular physical and mentioning it to your doctor.