Yes, Christians do believe that Jesus was not only fully human but also fully divine - that is, that He was God in human flesh. One reason is because of Jesus' own statements about Himself, such as His assertion that "I and the Father are one" (John 10:30).
How could Jesus die if He was God? One way to understand this is to realize that when Jesus' body died on the cross, He did not cease to exist. Yes, His body was dead–but His soul or spirit was still alive (just as ours will be when we die). Just moments before He died, we read that "Jesus called out with a loud voice, 'Father, into your hands I commit my spirit'" (Luke 23:46).
A far more important question about Jesus' death, however, is this: Why did He die? The Bible makes it clear that Jesus knew His enemies would try to put Him to death–and yet He refused to escape. Why? For one reason: Jesus came to become the final and complete sacrifice for our sins. We deserve to die for our sins, but He died in our place.
Why did He do this? He did it because God loves us and yearns for us to come to know Him and to love Him in return. He did it because He wants to forgive our sins and make us part of His family forever. Don't keep Christ at a distance any longer, but by faith open your heart to Him and commit your life to Him today.