I can't vouch for PCIE generations.
I have a seven year old Dell Inspiron 531. It has a PCIE-x16 slot and that was good enough to install an NVidea GTX750 Ti video card. The resulting transformation was dramatic. It is a high end card for the money. You should be able to get one for around £110. It also uses much less power than other cards.
That's something you have to consider. It's not just the graphics slot that you have that's important. I don't know the spec of your PC but if it's old and standard it may well have a very limited power supply unit (PSU) and it may not power your new fancy card...boom!
The GTX750 Ti will help here, because it only uses 60W but it will still draw more power than a built-in graphics chip. First of all find out exactly what graphics slot you have. Open up the PC and check it. Once you know what it is, you can then see if your chosen card will fit. If you are talking about altering the slot, you'll also need to buy another motherboard. That seems a bit mad, given that my old system accepts a very good card.
While you have the side panel off the PC, check your PSU for it's power rating. If its only 300W or so and a bit old,replace it. I'd recommend going up to at least 400W, but as I say, some graphics cards burn a lot more. NVidia recommend 300W minimum for the 750 Ti but that's with a new PSU. PSU's also get less efficient with age (don't we all) so don't bank on an old 300W unit coping. Personally, as I did, I'd get a new PSU. I got a Corsair 430W from Dabs.com along with the card. Cheap, prompt and no problems with the goods.
Hope this helps.
Feel free to ask me anything else. If I know, I'll help.