Study examples of great verse. Carroll's "Jabberwocky" is probably the most well-known nonsense poem, but check out Ogden Nash, Christopher Isherwood and E.E. Cummings. Even John Lennon wrote a nonsense poem.
Read your favorite nonsense poems out loud. Half of the joy in this verse comes from hearing the words strung together. Lear's limericks are especially enjoyable to read out loud to your children.
Buy a good rhyming dictionary or use Rhymezone. To combat the nonsensical nature of the words, nonsense verse gets its structure from effective rhymes.
Meditate on silly images. Even when the nonsense poem doesn't use made-up words or creatures, you want to incorporate unusual situations that stretch the mind of your readers.
Find a story to tell in your nonsense poem. While some verse sets up a silly situation, a longer poem sustains interest when the structure is recognizably a story. For instance, Carroll's "Jabberwocky" tells how "somebody killed something," as Alice put it.
Choose a time when you won't be disturbed to write your nonsense poem. Start out by making sense of the scene in your mind and, after you've gotten some lines down, go back and see how you can make them funny nonsense.
Share your poem with your friends. It should make them laugh in spite of the nonsense. Have them read it out loud to see if the rhymes and rhythm flow well.