hi there,
The sleeper train will probably be the cheapest and it takes 11 hours to get to Krakow. It travels via Warsaw and it's eight hours in total between
Berlin and Auschiwtz. On arrival at Krakow you then take a train or other transport to Auschwitcz which will take over an hour.
The sleeper train will set you off around $90. The bus
trip from Krakow to the Auschwitcz camp will be around $10