Riot is now adding the feature to let you change what champion shows up as your profile background to the League client. The new feature should make it on to the Public Beta Environment soon and then go live after some changes come through.
Not too much information is given in the PBE board post for the feature, but from what we can tell, it seems like you need to own the champion and skin in order to use said splash art as your background. The new feature is pretty self explanatory after all.
According to a tester in the comments, you will also be able to use the animated ultimate skin splashes, even if you don’t have the special icon equipped!
The feature needs some tweaks, changes, and lots of bug testing before it moves to the live servers. So go test it out and report back with changes you want to see!