Good computer

I've been playing LoL on a laptop for the past 2 years and I finally decided to upgrade to a desktop. I'm not familiar with what is a good computer for League so I was hoping you guys could help me pick out a good computer. My budget is about $350 and i know it isn't very high since my dad doesn't trust me with money (lol) but I've found two choices so far. My laptop is REALLY bad, it plays at 30-50 fps and goes down to 10 team fights (or sometimes less), so anything that can play league at a higher and stable fps would be great. I'm not sure exactly how to determine what makes a computer efficient but if you guys could point me in the right direction I would be grateful.

Here are the links to the two comps I've found so far: First Second

tldr: What's a good computer for gaming, mainly LoL, that fits my budget of $350

Thanks, Din