Playing the LicketySplit Banjo with fingerpicks can be a lot of fun, and can really bring out that bluegrass banjo flavor.
The basic fingerpicking pattern I use for most songs is the square roll. The square roll is a 4-count roll pattern that goes: Thumb-Index-Thumb-Middle.
To practice this roll, first pick the bass string with the thumb pick, then the middle string with the index finger, followed by the thumb on the bass string, and the middle finger on the treble string. (See part 1 of video below.)
When played at bluegrass speed, you will be playing 16th notes. You can count like this:
A variation of this pattern would be to alternate the string the thumb is playing: Thumb on the bass string, index finger on the middle string, thumb on the middle string, and middle finger on the treble string. (See part 2 of video below.)
Here is a demonstration of the two basic square rolls, followed by an example of Old Joe Clark: