League doesn't need anything too fancy to run at max graphics, so your concern with a graphics card will be if you want to stream other games. A $50-75 graphics card will likely do just fine for League.
For RAM, somewhere between 4GB and 8GB should do fine.
I run quad core Intel i5's on the computers I've built in an attempt to future proof them for other games. You could go with something cheaper for League and be all right, but I wouldn't go for the bottom of the barrel. After all, you do need to run your web browser, streaming software, and League at the same time.
I'd personally highly recommend a Solid State Drive. If your computer crashes, you're back up and in game in about 30 seconds instead of 5-10 minutes (less downtime for your stream as well). If you're looking to upload your content to YouTube, then I'd probably get a larger Hard Disk Drive as well. The ideal (although not set in stone) rule with this setup is that your OS and games go on the SSD, while the saved videos and your other documents go on the HDD. If I recall correctly, Windows 10 does a great job handling a new install on computers with both an SSD and HHD, but if I'm wrong you can search online how to set that up.
I personally recommend Open Broadcast Software for streaming if you're not already using it. It's free, and there are a few plugins that will automatically change your stream layout depending on which screen you're on (so one for Champ Select, one for Loading Screen, one for In Game, etc.)
