Riot Games' per-champion stat tracker, Eternals, is not a League of Legends addition some people like seeing whenever someone in the match hits a milestone. It can be annoying since it's basically advertising so here is how you turn it off.
Let's get into it immediately. It's not hard to turn Eternals off at all and the interface menu offers several options. Press Escape, click Interface on the left and scroll down to almost the bottom of the page. You will see Eternals interface options between Emotes and Combat Text sections.
The drop-down menu has all the Eternals enabled by default, meaning you will see milestones from both your and enemy team. It will probably be annoying to see the milestone enemy Syndra reached by chucking all her marbles at you so click the menu and you will be given two additional options - Self and Team Only or None.
Both are self-explanatory really. Self and Team Only will prevent the opposing team's Eternals from showing up on your screen while the None option will make all the stat trackers go away.
Considering you get this option, it's fair to say you can go pretty much unharmed into League of Legends matches since you will not be seeing the stat trackers and can completely avoid being bombarded by the notifications from other players which are basically glorified adverts.
Riot Games announced they would change the initial plans for Eternals, not long before this short guide was published. Therefore, the system will be changed so the Essence Emporium will offer some of these trackers twice a year for Blue Essence, the in-game currency.
You can read up on the planned changes on the official post but be aware that some fans suspected the initial announcement was just a ruse for the actual implementation plan where players would get outraged at the $1.750 price tag and latter settle down upon seeing a somewhat less egregious system. Those suspicions happened long before the announcement of the altered Eternals system.
On the bright side, you can just shut the notifications down entirely and completely ignore the Eternals' existence.