For your budget that CPU is fine though. If you've got more to spend, put it into a better graphics card (won't make a difference for LOL, but other games you will be able to dial up the settings).
But a 8320 & R7 265 is a well matched combination.
You should be able to get 60fps of LoL just fine, it's not a very demanding game at all.
If you're interested in the performance in other games, your best bet is to check out a few reviews of the R7 265. While your CPU isn't the best for gaming, it's enough to keep an R7 265 busy in just about any and every game, so you can expect to get very similar results to whatever they're posting in the video card reviews.
Tom's reviewed that card here: http://www.tomshardware.com/reviews/radeon-r7-265-review,3748.html
For your budget that CPU is fine though. If you've got more to spend, put it into a better graphics card (won't make a difference for LOL, but other games you will be able to dial up the settings).
But a 8320 & R7 265 is a well matched combination.