Less is More
Check your eyepiece to make sure that it really needs cleaning. Telescope optics are most often damaged during the cleaning process, because improper wiping and blowing drag dust particles across the lens. Dust has sharp edges, which scratch the surface of the optics. Cleaning solutions and tissues can also leave a residue which will cloud your view. If the eyepiece is only mildly dirty and doesn't interfere with your visual field, don't clean it.
Location
To clean an eyepiece properly, you need a clean, well-lit place. Wipe down the table where you will be cleaning. Make sure there is ample light, and use a lamp or an inspection light if needed. Lay out a microfiber cloth to set the eyepiece on, and only hold it with a microfiber cloth, since the oils from your fingers will leave a residue. Never clean your eyepiece in the field, as this exposes the lens to many materials that can scratch it, and you will probably not have the proper cleaning supplies or light on hand. Do not use red light either, or you will not be able to see what you are cleaning.
Blowing and Wiping
The best methods for removing dust and oils are to blow off the dust and wipe off the oils; however, you must use the proper tools. Never blow directly on the telescope eyepiece with your mouth. Droplets of moisture from your mouth will settle on the lens, causing spots and sticking to the dust. Instead, use a bulb-shaped puffer or a small syringe to blow the dust gently away. Hold the puffer at an angle, rather than blowing straight at the lens. Wipe away any oils and residues with a microfiber cloth or lens tissue, using a gentle circular motion. These items are available at camera stores. Do not use regular tissues, since these often contain lotions or other additives that will leave a residue. You can also brush dust off the lens with an antistatic brush.
Cleaning Solutions
There is much debate in the telescope world about the best cleaning solutions. Generally, any solution made for cleaning camera lenses or telescope lenses is safe to use. However, some solutions can leave a film. Reagent-grade acetone can be used, but not nail polish remover, as it contains perfumes. Read the labels carefully to make sure there are no ingredients in your cleaning solution besides alcohol and water. Use a cotton swab to gently rub the solution on the lens and loosen any grime or oil that is stuck on the lens. Mop up the alcohol with the dry end of the swab before the solution dries, or it will simply redeposit the oils on the eyepiece.