I have cleaned some very dirty lenses (acquired from careless friends) and this works every time.

You need:
a) a bottle of pure isopropyl alcohol
b) a box of good quality cotton buds of the type used for babies.

This is the procedure:
a) dip the clean cotton bud in the isopropyl alcohol, let the excess drain off.
b) apply the cotton bud at the center of the lens and gently wipe in a spiral motion that works towards the outer edge of the lens.
c) at the same time rotate the cotton bud so that a fresh clean surface is always presented to the lens. When you have rotated the cotton bud once discard it so that you don't re-use the dirty portion of the cotton bud.

In this way dirt is picked up by the wet cotton bud but never rubbed against the lens surface. In the same way you can gently polish off any drying marks that may form by using a dry cotton bud. Altogether you will use several cotton buds for just one lens surface.