You're using it to play league so no worries! As Mogs01gt was saying, building one is an option, and my personal recommendation as long as your not looking for a laptop. It's cheaper, you can upgrade afterwards very easily extending it's life as well as making it easy to add lights and other aftermarket goodies if you want later. Building a computer isn't as hard as it sounds, it's more of a grown-up version of square-peg square-hole once you know what each part does. That being said make sure you have a friend or family member who's built one before on hand to call for troubleshooting, that's when it has the possibility of being difficult.
If you don't want to do that then do some browsing and check out the specs. I can't suggest a specific one without your budget, what other games you'll be playing etc. but it's easy to find a good comp if you know what to look for.
---Processor: go for an i-series Intel of you can, it's what I use. But w/e brand higher Ghz usually means better.
---Graphics Card: I use Nvidia personally but that brand can get pricy fast. For league specifically you won't need an amazing one, but I suggest to get the best one you can so you can keep up with the times for longer. This also defines what video ports are coming out of your comp, HDMI, VGA or DVI for example.
---RAM: 6+Gbs will get most anything you need done. More is always better but don't go hogwild here.
These 3 things are what really affect your gameplay the most as far as hardware goes., but don't skimp on other parts like your hard drive for their sake, your computer will only be as strong as your weakest component.