The basic approach is to propel a solvent throught the system using compressed air. There are a couple of approaches to doing so.
The most elegant is to purchase a flush gun from a supplier such as acsource.com. Fill it with your preferred solvent - mineral spirits or lacquer thinner are two reasonable choices - pressurize it with compressed air, and blast the solvent through the condenser.
An alternative is to first introduce the solvent into the component being flushed using a suitable tool - a turkey baster or some such. Then use a blow gun attached to your air compressor to push the solvent through.
Lastly, you can purchase aerosol cans of flush from suppliers such as acsource.com. Again, you'll need compressed air to get all the flushing agent out of the component.
You don't need a big air compressor to do amateur a/c repair. A small 120v unit and blow gun can be purchased from Home Depot or similar stores for ~$150. Of course, something in that price range won't run any air tools, but it will fix your a/c and pump up your tires...