After you've found a good hand position, try a few exercises to get used to striking the strings in the clawhammer style. Clawhammer is very much an individualized approach to banjo playing. Even the best players play in a personalized way, doing what works best for them.
Although there are some general guidelines for you here, experiment with different ways of doing things to see what works best for you. Seek out other players for advice whenever you can, but do what sounds good and feels the most natural to you.
You can work up to playing the basic clawhammer strum by first getting comfortable striking individual notes. Use the right-hand index finger for playing melody notes (as indicated with the small letter i underneath each note in the banjo tab). However, the right-hand middle finger is an option preferred by many players. Try playing both ways and do what feels best to you.
Here are some tips for developing good right-hand clawhammer technique:
Having your finger knock the head after striking the 1st string is okay (the head is the round, white, top surface of the banjo). Remember, clawhammer is a percussive and rhythmic approach to playing the banjo. Don't be afraid to make some noise!
Most players use their index or middle fingernails to get a good clawhammer sound. In lieu of fingernails, you can use fingerpicks as extensions of the fingernails, but be sure to place the pick so that it covers the nail, not the pad of the finger as in bluegrass technique.
Some players, including Steve Martin, use artificial fingernails while others, like Riley Bagus, trim and shape regular plastic fingerpicks until they feel comfortable on the finger.
Try playing without picks first to see what kind of sound you get, and then play with picks or use artificial fingernails if you think yours aren't long enough to get enough contact with the strings or if you aren't getting a clear and forceful sound.
Remember that your finger is moving both down into the string and down toward the floor. Work toward developing a quick, decisive movement from the wrist that keeps your hand in control.
When your index or middle finger plays the 2nd, 3rd, or 4th string, it usually comes to rest against the next-highest string.
Some players bend their thumb at the joint to get that “claw” effect of hooking the finger underneath the strings. Other players extend the thumb outward. Again, try both and see which works best for you.