It is not random.
I have been champion only once and it was following being the "Champion" from the previous round, which meant getting the most kills, dealing the final blow, and being in the final squad.
I believe the marked champion is the player in the round who went the farthest in the previous round, generally from 60 players there is one who was the actual "Champion" (it announces that for you if you win). I am not sure what metric is used for tie breaking.
Keep in mind, the players in the current group with you compose the current round. None of them may have been in the squad which won in the previous round. It is possible as a result that the champion chosen was not actually the winner of a previous round, although unlikely, especially since the game has been around for quite some time now.
As for the kills, or the level, that is just on the banner, and is specific to that legend. So, if I have 1000 kills with Lifeline, but switch to using Bloodhound who I have only had 10 kills with, it will only show the kills I earned with Bloodhound (10). The level is always the same, but the display may change from banner to banner.