After a few minutes of playing any games, the screen freezes and I get a high-pitched whine. The situation has been steadily getting worse. For about a year we've got by through opening the case whenever games are being played.
I've checked all the fans. Upgraded to the current driver. Might I need to do some change to the BIOS? If so, how do I go about this?
The main PSU seems to get distinctly hot during game-playing.
I would suggest that either the power supply is not providing enough juice, especially since you say its running quite hot or the graphics card may have a problem. You mention something about opening the case. If this is because of heat then perhaps all you need to do is improve the cooling, perhaps by replacing a case fan or adding another. Worth mentioning here, if you are opening the case because of heat it can actually make things worse. This is because the idea is to have the air flow over the components and then out of the case, removing the sides can disrupt the airflow and make overheating worse.