Park Tool and others make a chain cleaner which clamps over the chain to clean it on the bike. These work fairly well when used with one of the citrus cleaners. You will probably need a few changes of cleaner until the chain comes clean. Be certain that the chain is dry before re-lubing it. A rinse with denatured alcohol will help remove the remaining water, as will heating the chain with a hair dryer.
When you replace the chain you can get one with a removable master link which can be opened to remove the chain without using a chain tool.