I believe in reincarnation although I am not really considered a Christian.
However, I do practice Christian Philosophy.
According to the Gospels, Jesus Christ did believe in reincarnation.
In fact according to the Gospels Jesus claims that John the Baptist is the Reincarnation of Elijah.
St. Mathew 17: 10-13
And the Disciples asked of Jesus,
"Why do the scribes say that Elijah must first come?"
And Jesus Answered,
"Elijah truly shall first come and restore all things."
"But I say unto you, that Elijah has already come, and they knew him not, but did to him what they wished." (cut off his head)
"Likewise, the son of man shall suffer of them."
Then the disciples knew that he spake unto them of John the Baptist.
Now, some of you Christians may say, well Elijah did not die, he was taken up into outer space in a UFO.
This is true, however, we have the account of the birth of John the Baptist (Christs cousin) in Luke Chapter 1.
So by this logic it is fair to assume that Elijah did die at some point, and his soul entered the body of "John the Baptist," as is pointed out by Jesus Christ in the afore mentioned passage.
That is, according to the Gospel.