Im flexible on the budget. Its an old computer, one that was assembled from individual components. In the location I'm in, we normally purchase the power supply, motherboard, hard drive, processor, video/audio card, etc individually and have someone assemble-- not to much in the way of branded computers here.
Do you know the wattage of the power supply?
If it's high enough, you may be able to buy a fast new video card without having to upgrade the power supply.
If you don't know the wattage of your computer's power supply, you can open the side of the case and check the label on the power supply.
For example on the top picture on
***this website*** you see a power supply and label, where the 650WTN tells you the power supply is 650 Watts. You shouldn't need to remove the power supply to check the label.
The power supply must have enough wattage to support whatever video card you choose.
The problem with Black Mirror 2 that people are having is that it requires shader 2.0 or higher. Many old video cards just don't have shader 2.0 functionality.
Besides the video card, check that your computer has enough RAM (system memory). For adventure games, 2 GB should be fine. But if you have less than that you might consider adding more.
How old is your computer? It's fine if you want a new one, but even a 6 year old computer will work with adventure games as long as the video card is good enough and you have enough RAM.