I worked on some of my lenses.
I typically start with a moist lens cloth. (not the paper kind but a real cloth - the type used for poly carbonate eye glasses) I use pure water first. Then go to a lens cleaning solution - very gentle type. I keep the pressure light and in circular motion. I have to fight the urge to apply pressure or use finger nails.... I repeat these process a few times.
After this, I use Q-tips and do the same.
If no joy at this point, it's either 1) forget it 2) risk damage or 3) send it to a pro.
It's so hard to fight the urge to apply pressure trying to clean the crud off.