After an epic round with Ggers and PatB3natar stabbing each other multiple times, it came to me that it would be nice to know what the PvP kill count was, the way BF3 does it? It would be especially welcome when we play as a clan, as it really helps to know how much you have destroyed a fellow clan mate

So, as well as having "{SAS}PatB3natar was knifed by Ggers" you could have following on the next line "That's 3-1 to {SAS}PatB3natar" so that you would know to hunt the player out, and objectives be damned

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While there isn't a way to deterimine how many times you have stabbed, shot, exploded, mortared, ran over, ran over with an APC ,there is a way to determine your total player skills. Look at the achievement thingy on the left side of the menu screen. There should be something like "Player Kills" there

At some point almost everything will be tracked (swimming distance, running distance, bullets shot, etc).
I absolutely agree that knowing how many times you killed a "human" would be the most exciting part.
It would be good if at least at the end of the round, you would have a scoreboard for the match you just had and not overall stats for all the games you already played.