Just FYI, your i5 3470S is not a particularly powerful processor. Any of the "S" or "T" versions have sacrificed some performance metrics in favor of saving power. Not necessarily saying it has anything to do with the proc, because we're talking about ping and on-line play here, but just keep that in mind.
You know issues like this can be especially hard to identify when you are talking about online play. There are a lot of variables here. Modem, router, wireless, wired, sharing connection...etc...
How does the game perform offline in single player mode? Is it smooth or laggy? Also do you have BF4 installed on a different harddrive than your other games, and is it an old school HDD or a newer SSD?
It would be interesting to see if you took your rig to a friends house and tried to play BF4 over their internet. See if it runs fine or does the same as it does at your home.
One thing to try as well although a pain, is to completely uninstall and then re-install the game. A pain I know, but you could possibly have a bad install or some bad files that cant be corrected by Origins file verification process. And as I asked earlier, if this game is on its own HDD it could be an issue with the drive. Not likely no, but I've seen stranger things, especially if its an old school slower HDD and not a newer/faster SSD.
Also - in regard to your upgrade -you do realize that the i7 6700k and the i5 3470s are two completely different socket types? You will need a new mother board as well.