If gaming in on the menu, and you have the right idea that LoL is the type of game that this computer can run, then I strongly recommend you stick with DDR5 equipped cards only.
Some ideas I'd suggest are: AMD HD 7750 or newer versions of it like the R7 250. On the Nvidia side, something like the GTX 650 or GT 730 DDR5. These are all small cards that use little power.
Which brings me to size. Your case, if 'slim', may only allow for low profile cards, single slot cards, or cards that are half length. I'd open the case to get a look at how much space you have to work with.