Take whatever transportation you need to get to Central Park West and 81st. I recommend the B or C train to 81st Street. Exit the subway on the north side of the train. Cross the street and enter the park. There will be a playground on your left and a tree in the middle of the path shortly after you enter the park. Go around the tree and follow the path to the right when the path branches. This will take you down a hill and up a hill and across the road that runs through the park. After crossing the street, continue up the hill and make a right in front of the big stone building (this is the Delacorte Theater). Go up the very uneven stone stairs at the end of the path. At the top of these stairs, look to your left. The Castle will be on the other side of a little clearing with a path running through it.
Alternatively, you can enter the park from the east side and 79th street. Enter the park on the north side of 79th street; keep going due west along the paths until you see the Lake and the Castle. Go around to your left. You'll eventually see the back entrance to the castle marked by more stone stairs; note that the first, narrow stairway you see going this way isn't the correct one; you need to keep going a few yards and you'll see some nice broad stairs, leading onto the landing; those are the ones you want. Go up these broad stairs to the castle.