You shouldnt have a problem running black ops 3 with those specs.
These are the recommended specs for BO3:
Operating System: Windows 7 64-Bit / Windows 8 64-Bit / Windows 8.1 64-Bit
Processor: Intel® Core™ i3-530 @ 2.93 GHz / AMD Phenom™ II X4 810 @ 2.60 GHz
Memory: 6 GB RAM
Graphics: NVIDIA® GeForce® GTX 470 @ 1GB / ATI® Radeon™ HD 6970 @ 1GB
DirectX: Version 11
Network: Broadband Internet connection
Sound Card: DirectX Compatible
Your PC meets all those requirements, is it your ping or frame rate that is bad? You said that you lag meaning you have a bad internet connection and should upgrade your internet and router, not PC components.
Good luck!