There isn't really a best way.
My CX300 is held together with a very thin RTV silicone adhesive. I can gently pry off the metal ring and separate the nozzle from the 'cup', and then soak the nozzle in a 5% solution of hydrogen peroxide (available at any pharmacy for extremely cheap).
the peroxide completely dissolves earwax and skin oils and will not damage the plastic or the screen (unless you leave it soaking for, say, two months).
But it turns out that many CX300's are held together with cyanoacrylates, or are ultrasonically welded, or something, and cannot be disassembled.
Someone once suggested using a toothpick or similar tool to push the screen down on one edge, allowing you to pull it out and clean it. Dunno what reinstalling it entails.