Your problem sounds like the same thing that's been happening to me. I'm almost positive it's your Internet.
I'm on DLS @ 5 Mbps, if I get more than one police helicopter on me the game starts to lag. If the cops are not on me I get 75-120+ FPS any where on the map. It"s only a problem online. In single player mod I'v never had a problem.
I know it's not my system as I'm running 2 EVGA 980's and a I7 3930k . If you have fast internet , I would check your firewall settings. Hope that helps ya out.