Is there any logic to who is the Game Champion?

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.

From my personal in game experience it seems to be whoever had the best standing the round before. Placement seems to be the thing that decides it but sometimes there will be multiple people who won from the last round where it then checks for Points (Respawns, Kills, Revives, ect.)

This link has an answer from someone backing up what I said but also saying that Distance Traveled and Weapon Accuracy might come into play.

There isn't any official source saying what determines the champion squad so the only way to get an answer would be to experiment in game but that would still be hard with there being 60 players per game with no way to tell how the champion did last round.