Here's the answer you probably don't want to hear. In most cases, it isn't possible to upgrade a laptop's graphics card. If you want better gaming performance, the only sensible option is to buy a new laptop.
The vast majority of laptops have integrated graphics, which means the GPU (graphics processing unit) is permanently attached to the motherboard, and not removable as it is in a desktop PC.
These days, a lot of CPUs contain the GPU, which means that you would have to replace the processor in order to upgrade the graphics. Taking Intel as an example, you'll find that the same or similar GPU is used as you work your way up the Core i7 range. That means that even if you do upgrade the processor, you won't get an improvement in graphics performance.
Plus, the moment you open up your laptop and start poking around at it's insides, you can say goodbye to your warranty.
If you're looking for a PC platform to play games on, getting a desktop is a much better deal, as you'll be spending less money and getting more performance. However, that's not an ideal solution for every one.
How to add a graphics card to a laptop
There is a way to add a graphics card to a laptop, but it's not for gaming purposes. Strange as it might sound, you can plug one in to a USB port. Doing this gives you an extra graphics output so you can drive an external monitor as well as your laptop's built-in screen. If your laptop already has a VGA, DisplayPort or HDMI output, adding a USB graphics card means you can drive a third screen: you're not limited to two displays.
For example, this StarTech adapter gives your laptop an HDMI output for around £50.