Step 2: Clean Bearings and Cups
Still using rag #1, try to clean out the bulk of the old grease from the bearing cups and the inside of the hub itself, after removing shields/seals. Most shields and seals will simply pry off, just be careful not to bend or kink them. If you see two holes across from each other on the seal carrier, it likely screws off in a counter-clockwise fashion. Twist the rag and thread it through the hub to remove the bulk of the grease. Feel free to use the brake cleaner solvent (after removing any seals!) to aid this process, but don't expect perfect cleanliness yet. Don't forget to clean the axle itself and the cone still fixed to it. Be very sure not to get any solvents on the tires.
If the bearings are in a cage, and you have additional ball bearings of the same gauge to make it a "loose ball" bearing, discard the cage and clean them all by soaking a portion of the rag in solvent and "shuffling" them in the rag by creating a pocket with your hand and rubbing the bearings into the rag.
By now, you should have a clean-looking set of parts. I hope that you have salvaged an old hub to make this a "loose-ball" bearing kit, but if not, be very sure to use the hem of the rag to clean out the cage. Remove balls from the cage by pressing them inward to push them out. Again, this is not clean yet, and if you plan to re-use the cage, you will have to follow additional steps.