Scripture tells us, "It is appointed that men die once and after death be judged" (Heb 10:27). Those in purgatory have been decisively judged after death and found to be in God’s friendship; otherwise they would be not in purgatory but in hell. The notion of reincarnation, by contrast, proposes that death brings not decisive judgment but simply endless opportunities to improve. Even the most wicked and unrepentant souls are simply reincarnated over and over until they finally come around and ascend to perfection. Thus, there is no "dying once" and no judgment after death.
Reincarnation is also incompatible with Christianity because it reduces the body to a disposable container. The person is essentially a spirit who inhabits one body or another. In Catholic belief, one’s own body is as uniquely one’s own and as essential to one’s nature as one’s spirit. Our destiny is resurrection, not reincarnation.