In reply to shaunparr1984:
Hi,you get a mini bus from in front of Zakopane bus/train station to Palenica/Morskie Oko.They start to take passengers from early in the morning but sometimes want to wait until the bus is full-you can negotiate to go with just you for a few extra zlotys.From Palenica it's a 9km walk/1.5hrs to Morskie Oko hut,where I would advise you to stay,as its in stunning location and once the day crowds are gone becomes the hub of Polish climbing.Then start early in the morning next day to get better morning weather and beat the crowds.It's an easy scramble to the top protected by chains in places.There could be still some snow in may in places.
It's doable in a single day with a very early start from Zakopane,say 4am
Hope this helps,other nice hikes in Morskie Oko area would be Mieguszowiecka Przelecz Pod Chlopkiem scramble or head to Dolina pieciu stawow across Szpiglasowa Przelecz(great views)