You can fly in on one of Peru's national airlines which are often going in and out of business - Star Peru is currently your only option in this category, or you could fly on Lan, a more stable South American airline. Lan is more reliable as far as timing and service, but StarPeru is currently the only airline that offers daytime flights to and from Pucallpa mid morning, and early evening, assuming that the flights are on time. Also, Lan has drastically different fares for Peruvian residents, and visitors can't get the reduced fares which require a Peruvian ID. StarPeru's fares, on the other hand, are the same for nationals and guests. You can book flights directly through the airlines, or you can use a Pucallpa-based travel agent, like Amazon World 011-51-61-57-5539. The advantage to using a travel agent my only experience is with Amazon World is that if a flight is delayed or canceled, they will go out of their way to notify you, or even to make arrangements if a cancellation causes you to miss another flight.
For the more adventurous, you could try coming in on a bus from Lima expect a two-day drive! Transmar runs direct buses from Lima.