How do I clean my baby's umbilical cord?

Depends. Some institutions use triple dye (purple) on the cord in an effort to avoid germs. This leatherizes the stump and makes separation take longer. Occasional cleaning may be needed to remove grime and thin soapy water or alcohol are fine. If the cord is allowed to air dry, occasional alcohol wipedowns accelerate drying. Bleeding at the base is common as the stump separates. Treat like you would a cut.
You don't have to. There's nothing you have to do to your baby's umbilical cord except keep it dry. People used to clean it with alcohol or betadine but it's best just left alone. Until it falls off you should not bathe your baby; you can give him a sponge bath but keep the umbilical stump dry. It'll fall off on its own at around 2 weeks. Then you can give him a bath.