The gang is correct! "Never" clean an AR from the muzzle to the chamber. Always at least pull the Take Down Pin remove the Bolt Carrier Assembly and clean from the Chamber to the Muzzle. As well as dissasemble the Bolt Carrier Group and clean it. In addition always clean the Barrel with the Front Sight or Gas Block in the upright position. If you clean it with the Front Sight or Gas Block facing the floor. You are dealing with first the hydrolic action of a wet patch with solvent in the bore. And also gravity will assist in getting the solvent down into your Gas Port and into the Gas Tube. Therefore possibly causing carbon build up in the Gas Port and Gas Tube since the solvent becomes fuel in a blast furnace when the weapon is fired. I am also with Primer1 and the others. Get a Dewey *22 CaL. Coated Rod or a Brass 22 Cal. Rod. Damaging the (lands and grooves) known as Rifleing or causing damage to the end of the Muzzle will cause serious accuracy problems for you. If you Google Dewey Cleaning Rods you will see who all handles them. Dewey, Brownells, Sinclair Int., Midway USA as mentioned all have great cleaning rods. I prefer the bearing handles on my rods. A lot of shops do not carry good rods! They handle the aluminum or steel rods! Also always use a Brass Cleaning Tip or Brass Jag Tip and Brush on the rod. They also make some good plastic type Tips and Brushes now also. But I prefer the Brass!
Purchase a good solvent to clean with like Shooters Choice, Hoppes #9, or other good solvent. Then wipe clean and use a good Lubricant on the parts. Your Bolt and Carrier should look slightly wet on the AR. We also have our good Mobile # Formula explained here on a FTF Thread or two, if you are interested. We use it at the LE Academy~ Great Lube for the AR-15 and M-16 Rifles. Have fun!
There are a lot of good folks here that are more than willing to assist if you need it!