I am having difficulties figuring out what kind of graphics card to get, or what to replace. My main reason for updating my machine is for better performance in games like League of Legends. This would require me to run my system at 60 frames per second. Currently I run probably 20 fps. Graphics settings are not an issue, price range is around $50 for a graphics card. Is this possible?
Rig as follows:
Processor: Intel(R) Pentium(R) CPU P6200 @ 2.13Hz 2.13Ghz
Ram: 4g 3.68 usable
99% of the time laptop's video card cannot be upgraded.
That's a bit of an exaggeration, but absolutely true that many laptops do not have a MXM slot or even a proprietary connection for an add-on graphics card. Since you stated your system uses integrated graphics, it's almost certain that - unless the card was previously removed for whatever reason - you do not have the connection interface for a new card.
Also, notebook graphics cards tend to cost drastically more than desktop cards, so I'm afraid $50 will not get you far. An inexpensive and old MXM card tends to run a good $250+.