Yes, you can take the display face glass off to clean. Be very careful here as it doesn't take much to break the glass or scratch the glass or LCD panel.
Before you start this job make sure you have plenty of desk space and have cleaned the area fully. Get a humidifier (running for a while before you start) to cut down on the static. Don't use any household cleaners as they will streak. A damp cloth with distilled water or reagent {very pure} quality of isopropyl or grain alcohol (don't use over the counter stuff!) is as far as you should go if you have something stubborn.
Before you clean the display use a can of can'ed air to blow out the system of all of the dust build up (moving is way from where your working on the display it's self)
Get plenty of 3M micro fiber poly dust cloths that don't shed to put down on the desk and use to wipe the glass. You should see about getting a good soft natural hair brush and vinyl gloves. You may want to wear a surgeons scrubs and hat so your body dirt & hair doesn't get on the glass as you work too (you don't want to take it apart again do you).
Heres a vid that shows how to remove the cover glass and clean between it and the display: Top Glass removal & cleaning
Heres a vid that goes deeper: Display disassembly
And of course the iFixIt guide:Screen removal guide
Some displays have metal tape wrapping around the edge. I found metal duct tape will work as a replacement if you damage it. Others have a Mylar tape which you can find at a good electronics supply house.
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