I know that the Kamikaze believed that their emporer was a God (like figure) and that this greatly influenced their actions. However I have no idea what they believed would happen to them after they'd become martyrs. I would love to know what they believed happened after death.
It would have depended on the individual of course, but it is probably not inaccurate to say that most believed they would become divine protectors of the Japanese Islands. The evidence for this comes from two sources. The term Kamikaze has some very important historic meanings to the Japanese people. The second and third largest aquatic invasion forces in all of history were sent by the Golden Horde to capture Japan. Both were destroyed just as they reached Japan by massive tsunamis. These tsunamis were called "kamikaze". Kaze is the Japanese word for wind. Kami are the divine Japanese spirits that are indigenous to the islands, and worshipped in Shinto. So by naming the suicide pilots "Kamikaze" the Japanese were referring back to this, so the pilots were seen as a recreation or modern version of the tsunamis. Next there is the shrine to the dead soldiers to consider. Located in Tokyo the Yasukuni Shrine was built to commemorate all the Japanese who died in wars between 1868 and 1947. All the people enshrined in this memorial are counted as Kami. I hope this helps! If anything was unclear let me know I will be happy to clarify as best I can.