This does sound very much like a CPU restriction, but BF4 is well known for its ability to use multiple cores effectively so maybe there's something else going on as well.
Try:
Run sweeps to clear out any possible malware and virus infections, you'll need both AV software like AVG or Avast! to clear out virus and Malwarebytes to clean out the malware infections, both offer free downloads.
Clean out the clutter. We all have it; Programs we don't use, programs that start during boot and we seldom, if ever use, anything running in the task bar will be using system resources, if you don't need it either uninstall it or set it not to run at startup. A good freebie to help keep routine clutter at bay is Ccleaner, even the basic, free version kills those pesky internet files like cookies and can knock out more than a few malware programs that reside in the Internet cache, just run it every time you log off the 'net or once a day.
Move any files or folders that are on the desktop into another location in the 'C' drive, Windows often loads them into main memory on startup and scans them for errors while the system is running, eating into CPU resources.
Make sure ALL the system drivers are up to date, sound, video, motherboard/chipset etc, sometimes older drivers don't play nice with newer ones.
Look into overclocking the current CPU, it's not something to just dive into,you can damage the CPU by getting heavy handed particularly with the voltage settings and it will require you to install a better CPU cooler but AMD parts do really benefit from a frequency boost.