It seems to me that number of kills and score is only recorded for the character who has done the most so for instance if you do 10,000 kills and then start a new character that new character would have to do 10,001 kills before it would show up on the leaderboard as a higher entry. However each time you restart a character although your position in the leaderboard does not change the number of kills does seem to reset to 0 on the weekly scoreboard.
In any event you can see the number of kills for your new character by looking at the stats. It is a shame that the leaderboard doesn't just show total score and total kills across all characters which would be a better reflection on your true position v other players. The scoreboards are totally fugged with stupid cheaters alas anyway.