(VLC is free and does the same as that advert...)
Generally you can pick up any nVidia or Ati based card from Amazon and get quite a jump from your on-board graphics. I assume the PC isn't ancient, so you are looking at a PCI Express graphics card.
Before you start - make sure you open up the PC case and check there is a slot available. (If you don't know what a PCI Express slot looks like, I can find you an image. It should be at the top of the stack of extra slots and look slightly longer and thinner)
Next we need to know if you have normal sized case or a slimline one.
As you want to watch video it make sense to get a lower end card with passive cooling (no noisy fans). Pointless spending out on a mad gaming card when all you need is HD.
As a random example of price ranges, here are some nVidia options as an example:
Amazon.co.uk: nvidia graphics PCI Express Graphics Cards
As you will notice, there are plenty in your price range. There is no point in a monster card for just video playback. You will see that the cards have names like 210, 640, 520, 440. The different generation of cards are signified by the first number in the hundreds column. So an 200 series card is the oldest tech, with 600 being much newer tech (and therefore more expensive). The second digit shows you how far up the range the card is - so a 440 is posher than a 420 while being based on the same generation of tech.
So it would be pointless for you to buy a 690 when a basic 420 would do the job. All you really need to do is check for that HDMI connector.
If I was just to grab something, I'd probably pick up this one:
Asus Nvidia GeForce GT 520 Silent Graphics Card (1GB DDR3, PCI Express 2.0, 700MHz/1333MHz, Low Profile, Nvidia PureHD Support): Amazon.co.uk: Computers & Accessories