At the moment of this post League of Legends is broken on Wine!
But it will probably get fixed in the next Wine Staging version (4.9) 
Here’s some tips to improve your experience on League:
1 > high polling rate mouse’s makes the FPS drop a lot when you move the cursor in-game, try using 125hz (even at 125hz it drops some frames, but is not that bad ), is a wine bug that affects other games too.
See: https://forum.winehq.org/viewtopic.php?f=8&t=32235
2 > the d9vk script should make the game perform better, if is not performing well try disabling that, on lutris click in “configure” and disable DXVK.
3 > try playing with med/low settings, disable shadows and all the fancy stuff.
4 > use Esync, enable that in the game settings on Lutris.
good luck.